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All courses Math III · A-APR.1 1 of 55
Add, subtract, and multiply polynomials beyond quadratic cases.

Add higher-degree polynomials by combining like terms

Problem
Find the sum of these higher-degree polynomials: (3x⁴ - 2x² + 5) + (x⁴ + 7x² - 9).
Your answer

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

Group the like terms

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

Since the two polynomials are being added, terms with the same variable part can be combined. The x⁴ terms go together, the x² terms go together, and the constants go together.

02

Combine the x^4 terms

\[3x^4~+~x^4~=~4x^4\]

Both terms are x⁴ terms, so add their coefficients: 3 + 1 = 4.

03

Combine the x^2 terms and constants

\[\begin{aligned} -2x^2~+~7x^2~=~5x^2 \\ 5~+~(-9)~=~-4 \end{aligned}\]

The x² terms combine because they have the same degree, and the constants combine because they have no variable. This gives 5x² and -4.

04

Write the final sum

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

Put the combined terms together in descending degree. The sum of the two polynomials is 4x⁴ + 5x² - 4.

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Another way

  1. You can also line the polynomials up vertically by degree and add each column: x⁴ terms, x² terms, and constants.

!

Common mistake

A common mistake is to add exponents or combine unlike terms, such as turning 3x⁴ + x⁴ into x⁸. When adding polynomials, the exponent stays the same and only the coefficients add.