Depth · one concept from first principles · step 2 of 4
Match coefficients of like terms
The equation holds for every value of . Find the value of . (, , and are constants.)
AExpressions
Your answer
How to solve
Strategy Convert to Algebra — Expand the left side, sort it by powers of x, and set each coefficient equal to the matching one on the right. That turns one identity into three small equations.
1STEP 1
Expand the left side
Distribute 3x over (Bx+5) and A over (Bx+5).
3x(Bx+5)+A(Bx+5)=3Bx²+15x+ABx+5A
Just the distributive property, twice.
A-APR.A.1Identify Subproblems2STEP 2
Sort by powers of x
Collect the two x terms together; x × x = x² gives the square term.
3Bx²+(15+AB)x+5A
Now each power of x appears once.
A-APR.A.1Organize Information In More Ways3STEP 3
Match the coefficients
Equal for every x means equal coefficients: x² with x², x with x, constant with constant.
3B=6, 15+AB=C, 5A=-10
Three equations come out of one.
6.EE.A.4Convert To Algebrafor every gives , , .
Why?
Collecting and into leaves one term for each power of on the left.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
Two polynomials equal for every have the same coefficient on each power, so each power gives one equation.
🧱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.
4STEP 4
Solve the small equations
The first and last give B and A at once; then the middle one gives C.
B=2, A=-2, C=15+(-2) × 2=11
Solve the easy ones first, then use them.
6.EE.A.4Convert To Algebra5STEP 5
Add them
Add the three constants.
A+B+C=-2+2+11=11
A and B cancel each other.
A-APR.A.1Identify SubproblemsAnswer
11
Put the values back: 3x(2x+5)-2(2x+5)=6x²+15x-4x-10=6x²+11x-10, which matches the right side with C=11.
Takeaway
If two polynomials are equal for every x, line up their terms and the unknowns fall out one by one.
- Expand the left side
- Sort by powers of x
- Match the coefficients
- Solve the small equations
- Add them
Where next?
Another one like thissuggested
Step back and solidifysuggested
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