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All courses Algebra II · A-APR.1 8 of 55
Add, subtract, and multiply polynomials beyond quadratic cases.

Add higher-degree polynomials by combining like terms

Problem
Simplify \((3x^4~-~2x^2~+~5)~+~(x^4~+~7x^2~-~9)\).
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Hint

Group the like terms before you add. Match the x⁴ terms together, the x² terms together, and the constants together.

Only like terms combine. Here, 3x⁴ + x⁴ = 4x⁴, -2x² + 7x² = 5x², and 5 + (-9) = -4.

Solution walkthrough

01

Identify the like-term groups

\[ax^n+bx^n=(a+b)x^n\]

Like terms have exactly the same variable part and exponent. When like terms are added, their coefficients add while the shared power stays unchanged. Here the x⁴ terms form one group, the x² terms form another, and the constants form a third.

02

Group this polynomial by degree

\[(3x^4+x^4)+(-2x^2+7x^2)+(5-9)\]

The plus sign between the original polynomials leaves every term's sign unchanged. The coefficient groups are therefore 3 and 1 for x⁴, -2 and 7 for x², and 5 and -9 for the constants.

03

Combine each coefficient group

\[(3+1)x^4+(-2+7)x^2+(5-9)=4x^4+5x^2-4\]

Add within each like-term group: 3+1=4, -2+7=5, and 5-9=-4. The exponents do not change because this is polynomial addition, not multiplication.

04

Check and state the simplified sum

\[4x^4~+~5x^2~-~4\]

For a check, substitute x=1. The original expression gives (3-2+5)+(1+7-9)=6+(-1)=5, and the simplified expression gives 4+5-4=5. The simplified sum is 4x⁴ + 5x² - 4, which matches choice A.

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Another way

  1. Write the polynomials vertically with the x⁴, x², and constant columns aligned, then add each column.

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Common mistake

Do not add exponents when adding like terms. The rule xm*xⁿ=xm⁺ⁿ is a multiplication rule; in 3x⁴+x⁴, only the coefficients add, giving 4x⁴.

Solution walkthrough video