Question2. Write the answer in the form $\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$ for $\frac{x^{3}-64}{x-4}$ and hence find $Q(x)$ and $R(x)$ .

Studdy Solution

STEP 1

1. The expression $\frac{x^3 - 64}{x - 4}$ is a rational function, where the numerator $x^3 - 64$ is a polynomial of degree 3 and the denominator $x - 4$ is a polynomial of degree 1.
2. We can perform polynomial division to rewrite the rational function in the form $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$ , where $Q(x)$ is the quotient and $R(x)$ is the remainder.
3. The polynomial $x^3 - 64$ can be factored or expanded using polynomial long division or synthetic division.

STEP 2

1. Perform polynomial long division of $x^3 - 64$ by $x - 4$ .
2. Identify the quotient $Q(x)$ from the division.
3. Identify the remainder $R(x)$ from the division.
4. Write the final expression in the form $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$ .

STEP 3

Set up the polynomial long division of $x^3 - 64$ by $x - 4$ .
$\begin{array}{r|rr} x - 4 & x^3 & 0x^2 & 0x & -64 \\ \hline & x^2 & 4x^2 & 0 & -64 \\ \end{array}$
Divide the leading term of the numerator by the leading term of the denominator: $x^3 \div x = x^2$ .

STEP 4

Multiply $x^2$ by $x - 4$ and subtract from the original polynomial: $x^3 - 4x^2 \\ \underline{- x^3 + 4x^2} \\ 0 + 4x^2 - 64$

STEP 5

Repeat the process with the new polynomial $4x^2 - 64$ : $\begin{array}{r|rr} x - 4 & x^2 & 4x & 16 \\ \hline & x^2 & 4x & 16 \\ \end{array}$
Divide the leading term of the new polynomial by the leading term of the denominator: $4x^2 \div x = 4x$ .

STEP 6

Multiply $4x$ by $x - 4$ and subtract from the new polynomial: $4x^2 - 16x \\ \underline{- 4x^2 + 16x} \\ 0 - 16x - 64$

STEP 7

Repeat the process with the new polynomial $-16x - 64$ : $\begin{array}{r|rr} x - 4 & x^2 & 4x & 16 \\ \hline & x^2 & 4x & 16 \\ \end{array}$
Divide the leading term of the new polynomial by the leading term of the denominator: $-16x \div x = -16$ .

STEP 8

Multiply $-16$ by $x - 4$ and subtract from the new polynomial: $-16x + 64 \\ \underline{- -16x + 64} \\ 0 + 0$

STEP 9

Identify the quotient $Q(x)$ from the division: $Q(x) = x^2 + 4x + 16$ .

STEP 10

Identify the remainder $R(x)$ from the division: $R(x) = 0$ .

STEP 11

Write the final expression in the form $\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$ : $\frac{x^3 - 64}{x - 4} = x^2 + 4x + 16 + \frac{0}{x - 4}$ Since the remainder is zero, the final expression simplifies to: $\frac{x^3 - 64}{x - 4} = x^2 + 4x + 16$
Therefore, $Q(x) = x^2 + 4x + 16$ and $R(x) = 0$ .

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