To find the width of the driveway, we need to divide the polynomial representing the area of the driveway by the polynomial representing the length of the driveway. Specifically, we will perform the polynomial division of [tex]\(5x^2 + 43x - 18\)[/tex] by [tex]\(x + 9\)[/tex].
Here are the steps:
1. Set up the division: We have to divide [tex]\(5x^2 + 43x - 18\)[/tex] by [tex]\(x + 9\)[/tex].
2. Divide the first term: Divide the leading term of the numerator [tex]\(5x^2\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex]. This gives us the first term in the quotient:
[tex]\[
\frac{5x^2}{x} = 5x
\][/tex]
3. Multiply the entire divisor by this term: Multiply [tex]\(x + 9\)[/tex] by [tex]\(5x\)[/tex] and subtract the result from the original polynomial:
[tex]\[
(x + 9) \cdot 5x = 5x^2 + 45x
\][/tex]
Subtracting this from [tex]\(5x^2 + 43x - 18\)[/tex] gives:
[tex]\[
(5x^2 + 43x - 18) - (5x^2 + 45x) = 43x - 45x - 18 = -2x - 18
\][/tex]
4. Repeat the process for the next term: Divide the leading term of the new polynomial [tex]\(-2x\)[/tex] by the leading term [tex]\(x\)[/tex]:
[tex]\[
\frac{-2x}{x} = -2
\][/tex]
5. Multiply the entire divisor by this term: Multiply [tex]\(x + 9\)[/tex] by [tex]\(-2\)[/tex] and subtract the result from [tex]\(-2x - 18\)[/tex]:
[tex]\[
(x + 9) \cdot (-2) = -2x - 18
\][/tex]
Subtracting this from [tex]\(-2x - 18\)[/tex] gives:
[tex]\[
(-2x - 18) - (-2x - 18) = 0
\][/tex]
After performing the division, our quotient (which represents the width of the driveway) is the polynomial obtained here:
[tex]\[
5x - 2
\][/tex]
Thus, the width of the driveway in terms of [tex]\( x \)[/tex] is:
[tex]\[
5x - 2
\][/tex]
