Question

4
enter the correct answer in the box.
use long division to rewrite the following expression.
\\(\frac{18x^2 + 5x + 5}{6x^2 - 4x + 1}\\)
write your answer in the format of \\( q(x) + \frac{r(x)}{b(x)} \\).

Explanation:

Step1: Divide the leading terms

Divide the leading term of the numerator \(18x^2\) by the leading term of the denominator \(6x^2\). So, \(\frac{18x^2}{6x^2} = 3\). This is the first term of the quotient \(q(x)\).

Step2: Multiply the divisor by 3

Multiply the entire divisor \(6x^2 - 4x + 1\) by 3: \(3(6x^2 - 4x + 1)=18x^2 - 12x + 3\).

Step3: Subtract this from the numerator

Subtract \(18x^2 - 12x + 3\) from the numerator \(18x^2 + 5x + 5\): \((18x^2 + 5x + 5)-(18x^2 - 12x + 3)=18x^2 + 5x + 5 - 18x^2 + 12x - 3 = 17x + 2\). This is the remainder \(r(x)\), and the divisor \(b(x)\) is still \(6x^2 - 4x + 1\).

Answer:

\(3+\frac{17x + 2}{6x^2 - 4x + 1}\)