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Grade 9 · lesson 2 of 4

Add Polynomials by Matching Powers

Polynomial addition combines coefficients of identical powers separately.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Add (2x²+3x−1)+(3x²−x+4). Candidate answers: (A) 5x²+2x+3 (B) 5x²+2x−5 (C) 5x²+4x+3 (D) 5x⁴+2x+3.
  • Independent check inputs: Add (3x²+3x−1)+(4x²−x+4). Candidate answers: (A) 7x²+2x+3 (B) 7x²+2x−5 (C) 7x²+4x+3 (D) 7x⁴+2x+3.

Learn the idea

Polynomials combine terms with nonnegative integer variable powers. Their degree describes the highest power, while addition, multiplication and factoring preserve equivalent expressions that can be checked by expansion or substitution. Polynomial addition combines coefficients of identical powers separately. Group quadratic, linear and constant terms to preserve each term’s variable structure while simplifying the whole sum.

Here is the method on the supplied teaching case. Combine terms with matching powers: (2+3)x²+(3−1)x+(−1+4)=5x²+2x+3.

Adding terms does not add their exponents. A missing power can be treated as a zero coefficient for organization; signs must stay attached to each original coefficient.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Add (2x²+3x−1)+(3x²−x+4). Candidate answers: (A) 5x²+2x+3 (B) 5x²+2x−5 (C) 5x²+4x+3 (D) 5x⁴+2x+3. Solution: Combine terms with matching powers: (2+3)x²+(3−1)x+(−1+4)=5x²+2x+3. The final answer is 5x²+2x+3. Adding terms does not add their exponents. A missing power can be treated as a zero coefficient for organization; signs must stay attached to each original coefficient.

Your turn

Check your reasoning

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Helpful tips
  • Polynomial addition combines coefficients of identical powers separately.
  • Adding terms does not add their exponents.
Add (3x²+3x−1)+(4x²−x+4).

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

7x²+2x+3

Combine terms with matching powers: (3+4)x²+(3−1)x+(−1+4)=7x²+2x+3.

Take it further

Changed-input task: Add (9x²+3x−1)+(10x²−x+4). Candidate answers: (A) 19x²+2x+3 (B) 19x²+2x−5 (C) 19x²+4x+3 (D) 19x⁴+2x+3. Show the mathematical reason, not only the selected answer. Expected result: 19x²+2x+3. Combine terms with matching powers: (9+10)x²+(3−1)x+(−1+4)=19x²+2x+3.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Add (4x²+3x−1)+(5x²−x+4).
Need a hint?
  • Polynomial addition combines coefficients of identical powers separately.
  • Adding terms does not add their exponents.
2. Add (5x²+3x−1)+(6x²−x+4).
Need a hint?
  • Polynomial addition combines coefficients of identical powers separately.
  • Adding terms does not add their exponents.
3. Add (6x²+3x−1)+(7x²−x+4).
Need a hint?
  • Polynomial addition combines coefficients of identical powers separately.
  • Adding terms does not add their exponents.

Your choices are temporary. Feedback appears after you check all the answers.

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