Distribute across Both Factors

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Expand (x+2)(x+3). Candidate answers: (A) 2x+5 (B) x²+5x+6 (C) x²+6 (D) x²+6x+5.
• Independent check inputs: Expand (x+3)(x+4). Candidate answers: (A) 2x+7 (B) x²+12 (C) x²+12x+7 (D) x²+7x+12.

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. Multiplying binomials requires multiplying each term of one factor by each term of the other. Four partial products can then be combined into a quadratic expression.

Here is the method on the supplied teaching case. Distribute each term: x²+3x+2x+6. Combining the middle terms gives x²+5x+6.

The middle terms usually do not vanish. Write all partial products before combining them; multiplying only first and last terms misses cross-products and changes the expression.

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.

Check your reasoning

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Take it further

Changed-input task: Expand (x+9)(x+10). Candidate answers: (A) 2x+19 (B) x²+19x+90 (C) x²+90 (D) x²+90x+19. Show the mathematical reason, not only the selected answer. Expected result: x²+19x+90. Distribute each term: x²+10x+9x+90. Combining the middle terms gives x²+19x+90.

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