Depth · one concept from first principles · step 2 of 4
Pair each monomial with a term by degree
and are monomials in , and . Simplify . (The degree of is greater than the degree of .)
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Clear the fraction and put the unknowns on one side. Two single-term unknowns must match the two terms on the other side, so list the two possible pairings and keep the one that fits the degree condition.
1STEP 1
Clear the fraction
Multiply both sides by 2x³.
A-8x⁴=(2x⁴+B) × 2x³=4x⁷+2x³B
No more fraction to worry about.
7.EE.A.2Convert To Algebra2STEP 2
Gather the unknowns
Move 2x³B to the left and 8x⁴ to the right.
A-2x³B=4x⁷+8x⁴
Unknowns on one side, known terms on the other.
6.EE.A.4Convert To Algebra3STEP 3
List the pairings
Each monomial on the left must take one whole term on the right, so there are only two ways to pair them.
(i) A=4x⁷, -2x³B=8x⁴ (ii) A=8x⁴, -2x³B=4x⁷
Only two cases to check.
8.EE.A.1Make A Systematic ListIn , one monomial must equal and the other .
Why?
The two sides are equal for every , so they carry the same coefficient on each power of .
🧱Equal polynomials match coefficient by coefficientTwo polynomials that agree for every input must have the same coefficient on every power, so one identity becomes a list of plain equations.
Why?
The right side needs two different powers, and , each with a nonzero coefficient, and each monomial on the left supplies only one power, so each monomial must match exactly one term.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
4STEP 4
Keep the case that fits
First case: B=8x⁴/(-2x³)=-4x, degree 1, smaller than 7. Second case: B=4x⁷/(-2x³)=-2x⁴, degree 4, the same as A, so it is ruled out.
A=4x⁷, B=-4x
The degree condition picks exactly one case.
8.EE.A.1Eliminate Possibilities5STEP 5
Divide
Divide the coefficients and subtract the exponents.
A/B=4x⁷/(-4x)=-x⁶
7-1=6.
8.EE.A.1Convert To AlgebraAnswer
-x⁶
Put A=4x⁷ back in: (4x⁷-8x⁴)/(2x³)=2x⁴-4x, which is 2x⁴+B with B=-4x.
Takeaway
A monomial is one term, so it can only match a term of its own degree.
- Clear the fraction
- Gather the unknowns
- List the pairings
- Keep the case that fits
- Divide
Where next?
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