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Question

Difficulty: MediumMultiplying Polynomials

Which of the following expressions is equivalent to the product of (5x−2)(5x - 2) and (x2−3x−4)(x^2 - 3x - 4)?

  1. A
    5x3+85x^3 + 8
  2. B
    5x3−17x2−26x−85x^3 - 17x^2 - 26x - 8
  3. 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8Answer
  4. D
    5x3−31x+85x^3 - 31x + 8

Answer

The correct answer is 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8.
The correct expression is found by systematically distributing each term in the first polynomial to each term in the second. First, multiplying 5x5x by (x2−3x−4)(x^2 - 3x - 4) yields 5x3−15x2−20x5x^3 - 15x^2 - 20x. Next, multiplying −2-2 by (x2−3x−4)(x^2 - 3x - 4) yields −2x2+6x+8-2x^2 + 6x + 8. Finally, combining the like terms (−15x2-15x^2 and −2x2-2x^2 becomes −17x2-17x^2; −20x-20x and 6x6x becomes −14x-14x) results in the simplified polynomial 5x3−17x2−14x+85x^3 - 17x^2 - 14x + 8.

Step-by-Step Solution

1
Distribute the first term of the binomial, 5x5x, to each term in the trinomial.
5x(x2−3x−4)=5x3−15x2−20x5x(x^2 - 3x - 4) = 5x^3 - 15x^2 - 20x
To expand the polynomials, every term in the first polynomial must be multiplied by every term in the second.
2
Distribute the second term of the binomial, −2-2, to each term in the trinomial.
−2(x2−3x−4)=−2x2+6x+8-2(x^2 - 3x - 4) = -2x^2 + 6x + 8
The second term, along with its negative sign, must also be multiplied by every term in the trinomial.
3
Add the results from Step 1 and Step 2 and combine like terms.
(5x3−15x2−20x)+(−2x2+6x+8)=5x3−17x2−14x+8(5x^3 - 15x^2 - 20x) + (-2x^2 + 6x + 8) = 5x^3 - 17x^2 - 14x + 8
Combining terms with the same variable to the same power fully simplifies the resulting polynomial.

Key Concept

Multiplying Polynomials using Distribution
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Topics

Multiplying Polynomials

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