Polynomial Multiplication

Grade 9 · algebra · 97 practice problems · read aloud

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What is Polynomial Multiplication? 🤔

Polynomial multiplication is the process of multiplying two or more polynomial expressions together. It's a fundamental skill in algebra that allows you to expand expressions, solve equations, and model real-world situations like calculating area. You'll use this to go from factored form, like (x + 2)(x + 3), to standard form, like x² + 5x + 6.

How to Multiply Polynomials: Step-by-Step

  1. Distribute Every Term: Multiply each term in the first polynomial by every term in the second polynomial.
  2. Use Exponent Rules: When multiplying variables, add the exponents (e.g., x² * x³ = x⁵).
  3. Combine Like Terms: After distributing, look for terms with the same variable and exponent and add their coefficients together.
  4. Write in Standard Form: Arrange your final answer from highest to lowest exponent.

Worked Examples

Example 1: Binomial × Binomial

Multiply: (x + 4)(x + 5)

  1. First: x * x = x²
  2. Outer: x * 5 = 5x
  3. Inner: 4 * x = 4x
  4. Last: 4 * 5 = 20
  5. Combine: x² + 5x + 4x + 20
  6. Final Answer: x² + 9x + 20

Example 2: Binomial × Trinomial

Multiply: (2x - 1)(x² + 3x - 2)

  1. 2x * x² = 2x³
  2. 2x * 3x = 6x²
  3. 2x * (-2) = -4x
  4. (-1) * x² = -x²
  5. (-1) * 3x = -3x
  6. (-1) * (-2) = 2
  7. Combine: 2x³ + 6x² - x² - 4x - 3x + 2
  8. Final Answer: 2x³ + 5x² - 7x + 2

Common Mistakes to Avoid 🚫

Forgetting to distribute to ALL terms: Don't just multiply the first terms! Use a systematic approach.

Incorrect exponent rules: Remember x * x = x² (adding exponents), not x² (multiplying exponents).

Missing negative signs: Be careful with signs when multiplying positive and negative terms.

Not combining all like terms: Double-check that you've collected every term that can be combined.

Tips & Tricks

FOIL Method: For two binomials, remember FOIL (First, Outer, Inner, Last).

Box Method: Draw a box to organize your work and ensure you multiply every term.

Check your work: Count the number of terms you should have before combining. A binomial × binomial should give you 4 terms before simplifying.

How to Practice

  • Start with simple binomials and work up to trinomials.
  • Create your own practice problems and check them.
  • Use online practice sites like Khan Academy for instant feedback.
  • Try "backwards" practice: factor a trinomial to see the connection.

Practice problems

6 of the 97, worked through step by step — try them before opening the answer.

1 (3x - 7)(5x + 9) = ?

Hint: Use the distributive property (FOIL method) to multiply each term in the first binomial by each term in the second binomial, then combine like terms.

Show the answer

Answer: 15x² - 8x - 63

  1. Multiply the First terms: 3x × 5x = 15x²
  2. Multiply the Outer terms: 3x × 9 = 27x
  3. Multiply the Inner terms: (-7) × 5x = -35x
  4. Multiply the Last terms: (-7) × 9 = -63
  5. Combine all terms: 15x² + 27x - 35x - 63
  6. Combine like terms: 15x² + (27x - 35x) - 63 = 15x² - 8x - 63

The answer is 15x² - 8x - 63.

2 (5x + 10)(3x - 5) = ?

Hint: Use the distributive property (FOIL method): multiply each term in the first binomial by each term in the second binomial, then combine like terms.

Show the answer

Answer: 15x² + 5x - 50

  1. Multiply the first terms: 5x × 3x = 15x²
  2. Multiply the outer terms: 5x × (-5) = -25x
  3. Multiply the inner terms: 10 × 3x = 30x
  4. Multiply the last terms: 10 × (-5) = -50
  5. Combine like terms: 15x² + (-25x + 30x) - 50 = 15x² + 5x - 50

The answer is 15x² + 5x - 50.

3 (3x² - 2x + 5)(x - 4) = ?

Hint: Apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial, then combine like terms.

Show the answer

Answer: 3x³ - 14x² + 13x - 20

  1. Multiply 3x² by each term in (x - 4)** 3x² * x = 3x³ 3x² * (-4) = -12x² So far: 3x³ - 12x² --- **
  2. Multiply -2x by each term in (x - 4)** -2x * x = -2x² -2x * (-4) = +8x Add these to what we have: 3x³ - 12x² - 2x² + 8x --- **
  3. Multiply +5 by each term in (x - 4)** 5 * x = 5x 5 * (-4) = -20 Add these: 3x³ - 12x² - 2x² + 8x + 5x - 20 --- **
  4. Combine like terms** x³ term: 3x³ (only one) x² terms: -12x² - 2x² = -14x² x terms: 8x + 5x = 13x Constant term: -20 --- **Final result:** 3x³ - 14x² + 13x - 20

Let's multiply (3x² - 2x + 5) by (x - 4) step by step. We will use the distributive property: multiply each term in the first polynomial by each term in the second polynomial, then combine like terms. --- **

4 (3x² - 2x + 5)(2x - 3) = ?

Hint: Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.

Show the answer

Answer: 6x³ - 13x² + 16x - 15

  1. Distribute 2x to each term in the trinomial** 2x * 3x² = 6x³ 2x * (-2x) = -4x² 2x * 5 = 10x So from 2x we get: 6x³ - 4x² + 10x --- **
  2. Distribute -3 to each term in the trinomial** -3 * 3x² = -9x² -3 * (-2x) = +6x -3 * 5 = -15 So from -3 we get: -9x² + 6x - 15 --- **
  3. Add the results from Step 1 and Step 2** From
  4. 6x³ - 4x² + 10x From
  5. - 9x² + 6x - 15 Now combine like terms: - x³ terms: 6x³ - x² terms: -4x² - 9x² = -13x² - x terms: 10x + 6x = 16x - constant terms: -15 So we have: 6x³ - 13x² + 16x - 15 --- **Final Answer:** 6x³ - 13x² + 16x - 15

Let's multiply the two expressions step by step. We have: (3x² - 2x + 5)(2x - 3) We will use the distributive property (also called FOIL for binomials, but here the first factor is a trinomial). --- **

5 (3x² - 2x + 5)(2x - 1) = ?

Hint: Apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial, then combine like terms.

Show the answer

Answer: 6x³ - 7x² + 12x - 5

  1. Distribute 2x to each term in the first polynomial. 2x * 3x² = 6x³ 2x * (-2x) = -4x² 2x * 5 = 10x So after distributing 2x, we have: 6x³ - 4x² + 10x
  2. Distribute -1 to each term in the first polynomial. (-1) * 3x² = -3x² (-1) * (-2x) = +2x (-1) * 5 = -5 So after distributing -1, we have: -3x² + 2x - 5
  3. Add the results from Step 1 and Step 2 together. From
  4. 6x³ - 4x² + 10x From
  5. - 3x² + 2x - 5 Now combine like terms: x³ terms: 6x³ x² terms: -4x² - 3x² = -7x² x terms: 10x + 2x = 12x Constant terms: -5
  6. Write the final polynomial. 6x³ - 7x² + 12x - 5 This matches the correct answer.

We are multiplying two polynomials: (3x² - 2x + 5) and (2x - 1). We can use the distributive property (also known as the FOIL method extended for more terms).

6 (4x² - 3x + 7)(2x + 5) = ?

Hint: Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.

Show the answer

Answer: 8x³ + 14x² - x + 35

  1. Multiply each term in the first polynomial by each term in the second polynomial: (4x²)(2x) = 8x³ (4x²)(5) = 20x² (-3x)(2x) = -6x² (-3x)(5) = -15x (7)(2x) = 14x (7)(5) = 35
  2. Write all terms: 8x³ + 20x² - 6x² - 15x + 14x + 35
  3. Combine like terms: 8x³ + (20x² - 6x²) = 8x³ + 14x² (-15x + 14x) = -x + 35
  4. Final answer: 8x³ + 14x² - x + 35
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