To solve the problem of dividing the polynomial [tex]\( \frac{8x^2 - 14x - 15}{2x - 7} \)[/tex], we need to perform polynomial long division. Let's go through this step by step.
1. Setup the Division:
We are dividing [tex]\( 8x^2 - 14x - 15 \)[/tex] by [tex]\( 2x - 7 \)[/tex].
2. First Term of the Quotient:
- Divide the leading term of the numerator, [tex]\( 8x^2 \)[/tex], by the leading term of the denominator, [tex]\( 2x \)[/tex].
- [tex]\( \frac{8x^2}{2x} = 4x \)[/tex].
- So, the first term in our quotient is [tex]\( 4x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( 4x \)[/tex] by the entire denominator [tex]\( 2x - 7 \)[/tex].
- [tex]\( 4x \cdot (2x - 7) = 8x^2 - 28x \)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(8x^2 - 14x - 15) - (8x^2 - 28x) = (-14x + 28x) - 15 = 14x - 15
\][/tex]
4. Next Term of the Quotient:
- Divide the new leading term of the remainder, [tex]\( 14x \)[/tex], by the leading term of the denominator, [tex]\( 2x \)[/tex].
- [tex]\( \frac{14x}{2x} = 7 \)[/tex].
- So, the next term in our quotient is [tex]\( + 7 \)[/tex].
5. Multiply and Subtract Again:
- Multiply [tex]\( 7 \)[/tex] by the entire denominator [tex]\( 2x - 7 \)[/tex].
- [tex]\( 7 \cdot (2x - 7) = 14x - 49 \)[/tex].
- Subtract this result from the remainder:
[tex]\[
(14x - 15) - (14x - 49) = -15 + 49 = 34
\][/tex]
- So, the remainder is [tex]\( 34 \)[/tex].
Putting it all together:
- The quotient is [tex]\( 4x + 7 \)[/tex].
- The remainder is [tex]\( 34 \)[/tex].
Thus, the result of dividing [tex]\( 8x^2 - 14x - 15 \)[/tex] by [tex]\( 2x - 7 \)[/tex] is:
[tex]\[ 4x + 7 + \frac{34}{2x - 7} \][/tex]
So, we have:
[tex]\[ \frac{8x^2 - 14x - 15}{2x - 7} = 4x + 7 + \frac{34}{2x - 7} \][/tex]
