Practice · Undo the wrong step to recover the polynomial · Sensim Math

Variant 1 easy

Someone was supposed to divide a polynomial by $2xy$ , but multiplied it by $2xy$ by mistake and got $20x^4y^2-12x^3y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2xy$ , but it was multiplied by $2xy$ instead, giving $20x^4y^2-12x^3y^3$ . Find what the correct division gives.

Givens

The mistaken result is $20x^4y^2-12x^3y^3$ .
The mistake multiplied by $2xy$ ; the correct step divides by $2xy$ .

Unknowns

The original polynomial divided by $2xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2xy

came out as the given result.

P \times 2xy = 20x^4y^2-12x^3y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2xy

undoes multiplying by

2xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{20x^4y^2}{2xy}-\frac{12x^3y^3}{2xy} = 10x^3y-6x^2y^2

x^4 \div x = x^3

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2xy

, again term by term (the distributive property).

(10x^3y-6x^2y^2) \div 2xy = 5x^2-3xy

Each term loses one more factor of

2xy

.

Answer:

5x^2-3xy

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2xy$ twice, that is by $4x^2y^2$ : $\frac{20x^4y^2}{4x^2y^2}=5x^2$ and $\frac{12x^3y^3}{4x^2y^2}=3xy$ . Same answer, $5x^2-3xy$ .

Another way: Skip the middle step: the mistaken result is $P \times 2xy$ , and the wanted result is $P \div 2xy$ , so divide the mistaken result by $(2xy)^2=4x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 2 easy

Someone was supposed to divide a polynomial by $2xy$ , but multiplied it by $2xy$ by mistake and got $28x^3y^4-20x^4y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2xy$ , but it was multiplied by $2xy$ instead, giving $28x^3y^4-20x^4y^3$ . Find what the correct division gives.

Givens

The mistaken result is $28x^3y^4-20x^4y^3$ .
The mistake multiplied by $2xy$ ; the correct step divides by $2xy$ .

Unknowns

The original polynomial divided by $2xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2xy

came out as the given result.

P \times 2xy = 28x^3y^4-20x^4y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2xy

undoes multiplying by

2xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{28x^3y^4}{2xy}-\frac{20x^4y^3}{2xy} = 14x^2y^3-10x^3y^2

x^3 \div x = x^2

and

y^4 \div y = y^3

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2xy

, again term by term (the distributive property).

(14x^2y^3-10x^3y^2) \div 2xy = 7xy^2-5x^2y

Each term loses one more factor of

2xy

.

Answer:

7xy^2-5x^2y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2xy$ twice, that is by $4x^2y^2$ : $\frac{28x^3y^4}{4x^2y^2}=7xy^2$ and $\frac{20x^4y^3}{4x^2y^2}=5x^2y$ . Same answer, $7xy^2-5x^2y$ .

Another way: Skip the middle step: the mistaken result is $P \times 2xy$ , and the wanted result is $P \div 2xy$ , so divide the mistaken result by $(2xy)^2=4x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 3 easy

Someone was supposed to divide a polynomial by $3xy$ , but multiplied it by $3xy$ by mistake and got $36x^3y^3-9x^2y^4$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $3xy$ , but it was multiplied by $3xy$ instead, giving $36x^3y^3-9x^2y^4$ . Find what the correct division gives.

Givens

The mistaken result is $36x^3y^3-9x^2y^4$ .
The mistake multiplied by $3xy$ ; the correct step divides by $3xy$ .

Unknowns

The original polynomial divided by $3xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

3xy

came out as the given result.

P \times 3xy = 36x^3y^3-9x^2y^4

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

3xy

undoes multiplying by

3xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{36x^3y^3}{3xy}-\frac{9x^2y^4}{3xy} = 12x^2y^2-3xy^3

x^3 \div x = x^2

and

y^3 \div y = y^2

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

3xy

, again term by term (the distributive property).

(12x^2y^2-3xy^3) \div 3xy = 4xy-y^2

Each term loses one more factor of

3xy

.

Answer:

4xy-y^2

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $3xy$ twice, that is by $9x^2y^2$ : $\frac{36x^3y^3}{9x^2y^2}=4xy$ and $\frac{9x^2y^4}{9x^2y^2}=y^2$ . Same answer, $4xy-y^2$ .

Another way: Skip the middle step: the mistaken result is $P \times 3xy$ , and the wanted result is $P \div 3xy$ , so divide the mistaken result by $(3xy)^2=9x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 4 easy

Someone was supposed to divide a polynomial by $2xy$ , but multiplied it by $2xy$ by mistake and got $36x^3y^2+40x^2y^4$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2xy$ , but it was multiplied by $2xy$ instead, giving $36x^3y^2+40x^2y^4$ . Find what the correct division gives.

Givens

The mistaken result is $36x^3y^2+40x^2y^4$ .
The mistake multiplied by $2xy$ ; the correct step divides by $2xy$ .

Unknowns

The original polynomial divided by $2xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2xy

came out as the given result.

P \times 2xy = 36x^3y^2+40x^2y^4

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2xy

undoes multiplying by

2xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{36x^3y^2}{2xy}+\frac{40x^2y^4}{2xy} = 18x^2y+20xy^3

x^3 \div x = x^2

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2xy

, again term by term (the distributive property).

(18x^2y+20xy^3) \div 2xy = 9x+10y^2

Each term loses one more factor of

2xy

.

Answer:

9x+10y^2

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2xy$ twice, that is by $4x^2y^2$ : $\frac{36x^3y^2}{4x^2y^2}=9x$ and $\frac{40x^2y^4}{4x^2y^2}=10y^2$ . Same answer, $9x+10y^2$ .

Another way: Skip the middle step: the mistaken result is $P \times 2xy$ , and the wanted result is $P \div 2xy$ , so divide the mistaken result by $(2xy)^2=4x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 5 medium

Someone was supposed to divide a polynomial by $3xy$ , but multiplied it by $3xy$ by mistake and got $18x^3y^2+45x^2y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $3xy$ , but it was multiplied by $3xy$ instead, giving $18x^3y^2+45x^2y^3$ . Find what the correct division gives.

Givens

The mistaken result is $18x^3y^2+45x^2y^3$ .
The mistake multiplied by $3xy$ ; the correct step divides by $3xy$ .

Unknowns

The original polynomial divided by $3xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

3xy

came out as the given result.

P \times 3xy = 18x^3y^2+45x^2y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

3xy

undoes multiplying by

3xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{18x^3y^2}{3xy}+\frac{45x^2y^3}{3xy} = 6x^2y+15xy^2

x^3 \div x = x^2

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

3xy

, again term by term (the distributive property).

(6x^2y+15xy^2) \div 3xy = 2x+5y

Each term loses one more factor of

3xy

.

Answer:

2x+5y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $3xy$ twice, that is by $9x^2y^2$ : $\frac{18x^3y^2}{9x^2y^2}=2x$ and $\frac{45x^2y^3}{9x^2y^2}=5y$ . Same answer, $2x+5y$ .

Another way: Skip the middle step: the mistaken result is $P \times 3xy$ , and the wanted result is $P \div 3xy$ , so divide the mistaken result by $(3xy)^2=9x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 6 medium

Someone was supposed to divide a polynomial by $4xy$ , but multiplied it by $4xy$ by mistake and got $32x^2y^3-48x^3y^2$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $4xy$ , but it was multiplied by $4xy$ instead, giving $32x^2y^3-48x^3y^2$ . Find what the correct division gives.

Givens

The mistaken result is $32x^2y^3-48x^3y^2$ .
The mistake multiplied by $4xy$ ; the correct step divides by $4xy$ .

Unknowns

The original polynomial divided by $4xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

4xy

came out as the given result.

P \times 4xy = 32x^2y^3-48x^3y^2

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

4xy

undoes multiplying by

4xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{32x^2y^3}{4xy}-\frac{48x^3y^2}{4xy} = 8xy^2-12x^2y

x^2 \div x = x

and

y^3 \div y = y^2

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

4xy

, again term by term (the distributive property).

(8xy^2-12x^2y) \div 4xy = 2y-3x

Each term loses one more factor of

4xy

.

Answer:

2y-3x

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $4xy$ twice, that is by $16x^2y^2$ : $\frac{32x^2y^3}{16x^2y^2}=2y$ and $\frac{48x^3y^2}{16x^2y^2}=3x$ . Same answer, $2y-3x$ .

Another way: Skip the middle step: the mistaken result is $P \times 4xy$ , and the wanted result is $P \div 4xy$ , so divide the mistaken result by $(4xy)^2=16x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 7 medium

Someone was supposed to divide a polynomial by $2xy$ , but multiplied it by $2xy$ by mistake and got $32x^3y^2-48x^2y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2xy$ , but it was multiplied by $2xy$ instead, giving $32x^3y^2-48x^2y^3$ . Find what the correct division gives.

Givens

The mistaken result is $32x^3y^2-48x^2y^3$ .
The mistake multiplied by $2xy$ ; the correct step divides by $2xy$ .

Unknowns

The original polynomial divided by $2xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2xy

came out as the given result.

P \times 2xy = 32x^3y^2-48x^2y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2xy

undoes multiplying by

2xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{32x^3y^2}{2xy}-\frac{48x^2y^3}{2xy} = 16x^2y-24xy^2

x^3 \div x = x^2

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2xy

, again term by term (the distributive property).

(16x^2y-24xy^2) \div 2xy = 8x-12y

Each term loses one more factor of

2xy

.

Answer:

8x-12y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2xy$ twice, that is by $4x^2y^2$ : $\frac{32x^3y^2}{4x^2y^2}=8x$ and $\frac{48x^2y^3}{4x^2y^2}=12y$ . Same answer, $8x-12y$ .

Another way: Skip the middle step: the mistaken result is $P \times 2xy$ , and the wanted result is $P \div 2xy$ , so divide the mistaken result by $(2xy)^2=4x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 8 medium

Someone was supposed to divide a polynomial by $3xy$ , but multiplied it by $3xy$ by mistake and got $54x^3y^3-18x^2y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $3xy$ , but it was multiplied by $3xy$ instead, giving $54x^3y^3-18x^2y^3$ . Find what the correct division gives.

Givens

The mistaken result is $54x^3y^3-18x^2y^3$ .
The mistake multiplied by $3xy$ ; the correct step divides by $3xy$ .

Unknowns

The original polynomial divided by $3xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

3xy

came out as the given result.

P \times 3xy = 54x^3y^3-18x^2y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

3xy

undoes multiplying by

3xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{54x^3y^3}{3xy}-\frac{18x^2y^3}{3xy} = 18x^2y^2-6xy^2

x^3 \div x = x^2

and

y^3 \div y = y^2

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

3xy

, again term by term (the distributive property).

(18x^2y^2-6xy^2) \div 3xy = 6xy-2y

Each term loses one more factor of

3xy

.

Answer:

6xy-2y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $3xy$ twice, that is by $9x^2y^2$ : $\frac{54x^3y^3}{9x^2y^2}=6xy$ and $\frac{18x^2y^3}{9x^2y^2}=2y$ . Same answer, $6xy-2y$ .

Another way: Skip the middle step: the mistaken result is $P \times 3xy$ , and the wanted result is $P \div 3xy$ , so divide the mistaken result by $(3xy)^2=9x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 9 hard

Someone was supposed to divide a polynomial by $5xy$ , but multiplied it by $5xy$ by mistake and got $25x^4y^2+75x^2y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $5xy$ , but it was multiplied by $5xy$ instead, giving $25x^4y^2+75x^2y^3$ . Find what the correct division gives.

Givens

The mistaken result is $25x^4y^2+75x^2y^3$ .
The mistake multiplied by $5xy$ ; the correct step divides by $5xy$ .

Unknowns

The original polynomial divided by $5xy$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

5xy

came out as the given result.

P \times 5xy = 25x^4y^2+75x^2y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

5xy

undoes multiplying by

5xy

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{25x^4y^2}{5xy}+\frac{75x^2y^3}{5xy} = 5x^3y+15xy^2

x^4 \div x = x^3

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

5xy

, again term by term (the distributive property).

(5x^3y+15xy^2) \div 5xy = x^2+3y

Each term loses one more factor of

5xy

.

Answer:

x^2+3y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $5xy$ twice, that is by $25x^2y^2$ : $\frac{25x^4y^2}{25x^2y^2}=x^2$ and $\frac{75x^2y^3}{25x^2y^2}=3y$ . Same answer, $x^2+3y$ .

Another way: Skip the middle step: the mistaken result is $P \times 5xy$ , and the wanted result is $P \div 5xy$ , so divide the mistaken result by $(5xy)^2=25x^2y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 10 hard

Someone was supposed to divide a polynomial by $2xy^2$ , but multiplied it by $2xy^2$ by mistake and got $24x^3y^4-4x^2y^5$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2xy^2$ , but it was multiplied by $2xy^2$ instead, giving $24x^3y^4-4x^2y^5$ . Find what the correct division gives.

Givens

The mistaken result is $24x^3y^4-4x^2y^5$ .
The mistake multiplied by $2xy^2$ ; the correct step divides by $2xy^2$ .

Unknowns

The original polynomial divided by $2xy^2$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2xy^2

came out as the given result.

P \times 2xy^2 = 24x^3y^4-4x^2y^5

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2xy^2

undoes multiplying by

2xy^2

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{24x^3y^4}{2xy^2}-\frac{4x^2y^5}{2xy^2} = 12x^2y^2-2xy^3

x^3 \div x = x^2

and

y^4 \div y^2 = y^2

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2xy^2

, again term by term (the distributive property).

(12x^2y^2-2xy^3) \div 2xy^2 = 6x-y

Each term loses one more factor of

2xy^2

.

Answer:

6x-y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2xy^2$ twice, that is by $4x^2y^4$ : $\frac{24x^3y^4}{4x^2y^4}=6x$ and $\frac{4x^2y^5}{4x^2y^4}=y$ . Same answer, $6x-y$ .

Another way: Skip the middle step: the mistaken result is $P \times 2xy^2$ , and the wanted result is $P \div 2xy^2$ , so divide the mistaken result by $(2xy^2)^2=4x^2y^4$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 11 hard

Someone was supposed to divide a polynomial by $2x^2y$ , but multiplied it by $2x^2y$ by mistake and got $12x^5y^2+28x^4y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $2x^2y$ , but it was multiplied by $2x^2y$ instead, giving $12x^5y^2+28x^4y^3$ . Find what the correct division gives.

Givens

The mistaken result is $12x^5y^2+28x^4y^3$ .
The mistake multiplied by $2x^2y$ ; the correct step divides by $2x^2y$ .

Unknowns

The original polynomial divided by $2x^2y$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

2x^2y

came out as the given result.

P \times 2x^2y = 12x^5y^2+28x^4y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

2x^2y

undoes multiplying by

2x^2y

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{12x^5y^2}{2x^2y}+\frac{28x^4y^3}{2x^2y} = 6x^3y+14x^2y^2

x^5 \div x^2 = x^3

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

2x^2y

, again term by term (the distributive property).

(6x^3y+14x^2y^2) \div 2x^2y = 3x+7y

Each term loses one more factor of

2x^2y

.

Answer:

3x+7y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $2x^2y$ twice, that is by $4x^4y^2$ : $\frac{12x^5y^2}{4x^4y^2}=3x$ and $\frac{28x^4y^3}{4x^4y^2}=7y$ . Same answer, $3x+7y$ .

Another way: Skip the middle step: the mistaken result is $P \times 2x^2y$ , and the wanted result is $P \div 2x^2y$ , so divide the mistaken result by $(2x^2y)^2=4x^4y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Variant 12 hard

Someone was supposed to divide a polynomial by $3x^2y$ , but multiplied it by $3x^2y$ by mistake and got $9x^5y^2-36x^4y^3$ . What is the result of the correct calculation?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A polynomial should have been divided by $3x^2y$ , but it was multiplied by $3x^2y$ instead, giving $9x^5y^2-36x^4y^3$ . Find what the correct division gives.

Givens

The mistaken result is $9x^5y^2-36x^4y^3$ .
The mistake multiplied by $3x^2y$ ; the correct step divides by $3x^2y$ .

Unknowns

The original polynomial divided by $3x^2y$ .

Constraints

The original polynomial is not given, so it has to be recovered from the mistake first.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra#7 Identify Subproblems

The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

3 · Execute3 carry out the plan

1Name the lost polynomial

#13 Convert to Algebra 7.EE.B.4

Call the original polynomial

P

. The mistake says

P

times

3x^2y

came out as the given result.

P \times 3x^2y = 9x^5y^2-36x^4y^3

One equation holds the only unknown.

2Undo the multiplication

#11 Work Backwards 8.EE.A.1

Dividing by

3x^2y

undoes multiplying by

3x^2y

. Divide each term separately, subtracting exponents of matching letters.

P = \frac{9x^5y^2}{3x^2y}-\frac{36x^4y^3}{3x^2y} = 3x^3y-12x^2y^2

x^5 \div x^2 = x^3

and

y^2 \div y = y

are the exponent laws.

3Do the intended division

#7 Identify Subproblems 6.EE.A.3

Now divide the recovered polynomial by

3x^2y

, again term by term (the distributive property).

(3x^3y-12x^2y^2) \div 3x^2y = x-4y

Each term loses one more factor of

3x^2y

.

Answer:

x-4y

4 · Reviewdoes it hold up?

Going from the mistaken result to the right one means dividing by $3x^2y$ twice, that is by $9x^4y^2$ : $\frac{9x^5y^2}{9x^4y^2}=x$ and $\frac{36x^4y^3}{9x^4y^2}=4y$ . Same answer, $x-4y$ .

Another way: Skip the middle step: the mistaken result is $P \times 3x^2y$ , and the wanted result is $P \div 3x^2y$ , so divide the mistaken result by $(3x^2y)^2=9x^4y^2$ in one go.

Standardsmin grade 8

8.EE.A.1 Know and apply the properties of integer exponents — Dividing powers of $x$ and $y$ by subtracting exponents.
6.EE.A.3 Apply properties to generate equivalent expressions — Dividing a polynomial by a monomial term by term.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the mistake as an equation with an unknown polynomial.

💡Takeaway. When a step was done the wrong way, undo it first, then do it the right way.

Undo the wrong step to recover the polynomial · 12 practice problems

Generated variants — 12

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