Question

question
use the long division method to find the result when $-9x^3 + 4x + 9$ is divided by $3x - 4$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Arrange the polynomial

First, write the dividend \(-9x^3 + 0x^2 + 4x + 9\) (adding the missing \(x^2\) term with coefficient 0) and the divisor \(3x - 4\).

Step2: Divide the leading terms

Divide the leading term of the dividend \(-9x^3\) by the leading term of the divisor \(3x\). So, \(\frac{-9x^3}{3x}=-3x^2\). This is the first term of the quotient.

Step3: Multiply and subtract

Multiply the divisor \(3x - 4\) by \(-3x^2\): \(-3x^2(3x - 4)=-9x^3 + 12x^2\). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step4: Repeat the process

Now, divide the leading term of the new dividend \(-12x^2\) by the leading term of the divisor \(3x\). \(\frac{-12x^2}{3x}=-4x\). This is the next term of the quotient.

Step5: Multiply and subtract again

Multiply the divisor \(3x - 4\) by \(-4x\): \(-4x(3x - 4)=-12x^2 + 16x\). Subtract this from the new dividend:

$$
LATEXBLOCK1
$$

Step6: Repeat once more

Divide the leading term of the new dividend \(-12x\) by the leading term of the divisor \(3x\). \(\frac{-12x}{3x}=-4\). This is the next term of the quotient.

Step7: Multiply and subtract the last time

Multiply the divisor \(3x - 4\) by \(-4\): \(-4(3x - 4)=-12x + 16\). Subtract this from the new dividend:

$$
LATEXBLOCK2
$$

Step8: Write the final form

The quotient \(q(x)\) is \(-3x^2 - 4x - 4\) and the remainder \(r(x)\) is \(-7\), and the divisor \(b(x)\) is \(3x - 4\). So the result is \(q(x)+\frac{r(x)}{b(x)}=-3x^2 - 4x - 4+\frac{-7}{3x - 4}\) or simplified as \(-3x^2 - 4x - 4-\frac{7}{3x - 4}\).

Answer:

\(-3x^2 - 4x - 4-\dfrac{7}{3x - 4}\)