Question

mary wants to find an expression she can use to determine how fast she has traveled. she knows that her distance is represented by (-24x³ - 70x² - 59x - 15) and that her time is represented by (-4x - 3). which expression, in simplest terms, represents her rate?
a -12x² - 29x - 15
b -8x² + 10x - 3
c 6x² + 13x + 5
d 10x² + 16x + 3

Explanation:

Step1: Recall the formula for rate

Rate \( r \) is given by the formula \( r=\frac{\text{distance}(d)}{\text{time}(t)} \). So we need to divide the polynomial \( -24x^{3}-70x^{2}-59x - 15 \) by \( - 4x-3 \).

Step2: Perform polynomial long division

We divide \( -24x^{3}-70x^{2}-59x - 15 \) by \( -4x - 3 \).

• First term: \( \frac{-24x^{3}}{-4x}=6x^{2} \). Multiply \( -4x - 3 \) by \( 6x^{2} \): \( 6x^{2}(-4x - 3)=-24x^{3}-18x^{2} \). Subtract this from the dividend:

\( (-24x^{3}-70x^{2}-59x - 15)-(-24x^{3}-18x^{2})=-24x^{3}-70x^{2}-59x - 15 + 24x^{3}+18x^{2}=-52x^{2}-59x - 15 \)

• Second term: \( \frac{-52x^{2}}{-4x} = 13x \). Multiply \( -4x - 3 \) by \( 13x \): \( 13x(-4x - 3)=-52x^{2}-39x \). Subtract this from the new dividend:

\( (-52x^{2}-59x - 15)-(-52x^{2}-39x)=-52x^{2}-59x - 15 + 52x^{2}+39x=-20x - 15 \)

• Third term: \( \frac{-20x}{-4x}=5 \). Multiply \( -4x - 3 \) by \( 5 \): \( 5(-4x - 3)=-20x - 15 \). Subtract this from the new dividend:

\( (-20x - 15)-(-20x - 15)=-20x - 15 + 20x + 15 = 0 \)

So the result of the division is \( 6x^{2}+13x + 5 \).

Answer:

C. \( 6x^{2}+13x + 5 \)