To divide the polynomial [tex]\( P(x) = 8x^2 - 3x - 17 \)[/tex] by [tex]\( D(x) = 4x - 1 \)[/tex], we can use polynomial long division. Our goal is to express the quotient in the form:
[tex]\[
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
\][/tex]
where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R(x) \)[/tex] is the remainder.
Let's go through the steps in polynomial long division:
1. Setup: Write the division as:
[tex]\[
\frac{8x^2 - 3x - 17}{4x - 1}
\][/tex]
2. First Division:
- Divide the leading term of the numerator [tex]\( 8x^2 \)[/tex] by the leading term of the denominator [tex]\( 4x \)[/tex]. This gives:
[tex]\[
\frac{8x^2}{4x} = 2x
\][/tex]
- Multiply the entire divisor by [tex]\( 2x \)[/tex]:
[tex]\[
(4x - 1) \cdot 2x = 8x^2 - 2x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(8x^2 - 3x - 17) - (8x^2 - 2x) = -x - 17
\][/tex]
3. Second Division:
- Divide the new leading term [tex]\( -x \)[/tex] by the leading term of the divisor [tex]\( 4x \)[/tex]:
[tex]\[
\frac{-x}{4x} = -\frac{1}{4}
\][/tex]
- Multiply the divisor by [tex]\(-\frac{1}{4}\)[/tex]:
[tex]\[
(4x - 1) \cdot -\frac{1}{4} = -x + \frac{1}{4}
\][/tex]
- Subtract this result from the polynomial:
[tex]\[
(-x - 17) - (-x + \frac{1}{4}) = -17 - \frac{1}{4} = -\frac{68}{4} - \frac{1}{4} = -\frac{69}{4}
\][/tex]
So, our quotient ([tex]\( Q(x) \)[/tex]) and remainder ([tex]\( R(x) \)[/tex]) are:
[tex]\[
Q(x) = 2x - \frac{1}{4}, \quad R(x) = -\frac{69}{4}
\][/tex]
Therefore, we can write the division as:
[tex]\[
\frac{P(x)}{D(x)} = 2x - \frac{1}{4} + \frac{-\frac{69}{4}}{4x - 1}
\][/tex]
In a more simplified form, it will be:
[tex]\[
\frac{P(x)}{D(x)} = 2x - \frac{1}{4} + \frac{-69}{4(4x - 1)}
\][/tex]
