To find the width of the model in terms of [tex]\( x \)[/tex], we need to divide the area of the base by the length. The given area is [tex]\( 12x^2 - 11x - 5 \)[/tex] and the length is [tex]\( 3x + 1 \)[/tex].
We start by performing polynomial division of [tex]\( 12x^2 - 11x - 5 \)[/tex] by [tex]\( 3x + 1 \)[/tex].
1. Divide the leading term of the numerator ([tex]\( 12x^2 \)[/tex]) by the leading term of the denominator ([tex]\( 3x \)[/tex]):
[tex]\[
\frac{12x^2}{3x} = 4x
\][/tex]
So, the first term of the quotient is [tex]\( 4x \)[/tex].
2. Multiply [tex]\( 4x \)[/tex] by the entire divisor [tex]\( 3x + 1 \)[/tex]:
[tex]\[
4x \cdot (3x + 1) = 12x^2 + 4x
\][/tex]
3. Subtract this result from the original polynomial:
[tex]\[
(12x^2 - 11x - 5) - (12x^2 + 4x) = -15x - 5
\][/tex]
4. Now, divide the new leading term of the remainder ([tex]\( -15x \)[/tex]) by the leading term of the divisor ([tex]\( 3x \)[/tex]):
[tex]\[
\frac{-15x}{3x} = -5
\][/tex]
So, the next term of the quotient is [tex]\( -5 \)[/tex].
5. Multiply [tex]\( -5 \)[/tex] by the entire divisor [tex]\( 3x + 1 \)[/tex]:
[tex]\[
-5 \cdot (3x + 1) = -15x - 5
\][/tex]
6. Subtract this result from the current remainder:
[tex]\[
(-15x - 5) - (-15x - 5) = 0
\][/tex]
So, the quotient, which is the width, is:
[tex]\[
4x - 5
\][/tex]
Therefore, the width of the model in terms of [tex]\( x \)[/tex] is [tex]\( 4x - 5 \)[/tex].
