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Difficulty: HardMultiplying Polynomials

A civil engineer is designing a rectangular water reservoir. The length of the reservoir is represented by the polynomial (4x−3)(4x - 3) meters, and the cross-sectional area of the side wall is represented by (2x2+5x−1)(2x^2 + 5x - 1) square meters. If the total volume of the reservoir is calculated by multiplying its length by its cross-sectional area, which of the following polynomials represents the total volume in cubic meters?

  1. 8x3+14x2−19x+38x^3 + 14x^2 - 19x + 3Answer
  2. B
    8x3+14x2−19x−38x^3 + 14x^2 - 19x - 3
  3. C
    8x3+20x2−4x−38x^3 + 20x^2 - 4x - 3
  4. D
    8x3+16x2−21x+38x^3 + 16x^2 - 21x + 3

Answer

The correct polynomial representing the volume is 8x3+14x2−19x+38x^3 + 14x^2 - 19x + 3.
The correct expression is found by properly distributing both terms of the binomial (4x−3)(4x - 3) to all three terms of the trinomial (2x2+5x−1)(2x^2 + 5x - 1), maintaining correct signs, and accurately combining the resulting like terms.

Step-by-Step Solution

1
Set up the multiplication of the two polynomials.
V=(4x−3)(2x2+5x−1)V = (4x - 3)(2x^2 + 5x - 1)
The problem states that the volume is calculated by multiplying the length by the cross-sectional area.
2
Distribute the first term of the binomial, 4x4x, to every term in the trinomial.
4x(2x2)+4x(5x)+4x(−1)=8x3+20x2−4x4x(2x^2) + 4x(5x) + 4x(-1) = 8x^3 + 20x^2 - 4x
The distributive property requires multiplying every term in the second polynomial by the first term of the first polynomial.
3
Distribute the second term of the binomial, −3-3, to every term in the trinomial.
−3(2x2)−3(5x)−3(−1)=−6x2−15x+3-3(2x^2) - 3(5x) - 3(-1) = -6x^2 - 15x + 3
Carefully apply the negative sign to all terms during multiplication, noting that a negative times a negative yields a positive.
4
Combine the results and group the like terms.
8x3+(20x2−6x2)+(−4x−15x)+38x^3 + (20x^2 - 6x^2) + (-4x - 15x) + 3
Like terms (terms with the exact same variable configuration) must be grouped together to simplify the expression.
5
Perform the final addition and subtraction to simplify completely.
8x3+14x2−19x+38x^3 + 14x^2 - 19x + 3
This yields the fully simplified polynomial in standard form.

Key Concept

Multiplying a binomial by a trinomial using the Distributive Property
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Topics

Multiplying Polynomials

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