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All courses Math II · A-APR.1 1 of 73
Add, subtract, and multiply polynomials; understand polynomial closure under these operations.

Add polynomials and write the sum in standard form

Problem
Add the polynomials (3x² + 5x - 7) and (2x² - x + 4), and write their sum in standard form. Then Choose the option showing the three coefficients of the sum in descending-degree order (quadratic, linear, constant), separated by commas. Include 0 for any missing degree; do not enter x or words.
Your answer

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

Group the like terms

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

The terms \(3x^2\) and \(2x^2\) share the same variable part, as do \(5x\) and \(-x\). The constants form the third group.

02

Add the signed coefficients

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

Keep each exponent and add within its coefficient group: \(3 + 2 = 5\), \(5 - 1 = 4\), and \(-7 + 4 = -3\).

03

Write standard form and the coefficient list

\[\begin{aligned} 5x^2~+~4x~-~3 \\ 5,~4,~-3 \end{aligned}\]

The polynomial is already ordered from \(x^2\) to \(x\) to the constant term. Therefore, its requested coefficient list is \(5, 4, -3\).

04

Verify by substitution

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

At \(x = 1\), the original sum and the simplified polynomial both equal \(6\). This substitution check supports the result.

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Another way

  1. Use coefficient lists instead: \((3, 5, -7) + (2, -1, 4) = (5, 4, -3)\). Each position represents one matching power of \(x\).

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Common mistake

Do not add exponents when adding like terms. In \(3x^2 + 2x^2\), the exponent remains \(2\), while the coefficients add to \(5\).