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All courses Algebra I · A-APR.1 47 of 76
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.

Add polynomials and write the sum in standard form

Problem
Add the polynomials \((3x^2~+~5x~-~7)\) and \((2x^2~-~x~+~4)\). Give the coefficients of the sum in descending-degree order, including zeros for missing degrees.
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Hint

Group terms with the same power of \(x\): combine the two \(x^2\) terms, then the two \(x\) terms, then the constants.

When adding like terms, keep the variable and exponent unchanged and add the signed coefficients. A power that does not appear has coefficient \(0\).

Solution walkthrough

01

Align terms by degree

\[\begin{aligned} (3x^2~+~5x~-~7)~+~(2x^2~-~x~+~4) \\ =~(3+2)x^2~+~(5+(-1))x~+~(-7+4) \end{aligned}\]

Like terms have the same variable raised to the same power. The quadratic coefficients are 3 and 2, the linear coefficients are 5 and -1 because -x=-1x, and the constants are -7 and 4.

02

Add the signed coefficients

\[\begin{aligned} (3+2)x^2~+~(5+(-1))x~+~(-7+4) \\ =~5x^2~+~4x~-~3 \end{aligned}\]

Adding within each degree gives 3+2=5, 5+(-1)=4, and -7+4=-3. The powers of x stay unchanged because this is addition, not multiplication.

03

Report and check the coefficients

\[\begin{aligned} 5x^2~+~4x~-~3~->~5,~4,~-3 \\ x=1:~(3+5-7)+(2-1+4)=6 \\ 5+4-3=6 \end{aligned}\]

The sum is already in descending-degree order, so the requested coefficients are 5, 4, -3. As a check, both the original sum and the simplified polynomial equal 6 when x=1.

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Another way

  1. Represent the polynomials by aligned coefficient lists: (3,5,-7)+(2,-1,4)=(5,4,-3).

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Common mistake

Do not add the exponents of like terms. Adding 3x² and 2x² adds only the coefficients, producing 5x² rather than 5x⁴.

Solution walkthrough video