Let's perform polynomial long division for the given problem step-by-step.
We want to divide [tex]\( 8x^3 + 26x^2 - 25x - 25 \)[/tex] by [tex]\( 4x - 5 \)[/tex].
1. First Division:
- Divide the leading term of the dividend [tex]\( 8x^3 \)[/tex] by the leading term of the divisor [tex]\( 4x \)[/tex]:
[tex]\[
\frac{8x^3}{4x} = 2x^2
\][/tex]
- Multiply the entire divisor [tex]\( 4x - 5 \)[/tex] by [tex]\( 2x^2 \)[/tex] and subtract the result from the original dividend:
[tex]\[
(8x^3 + 26x^2 - 25x - 25) - (2x^2 \cdot (4x - 5)) = (8x^3 + 26x^2 - 25x - 25) - (8x^3 - 10x^2)
\][/tex]
- Simplify:
[tex]\[
8x^3 + 26x^2 - 25x - 25 - 8x^3 + 10x^2 = 36x^2 - 25x - 25
\][/tex]
2. Second Division:
- Divide the leading term of the new dividend [tex]\( 36x^2 \)[/tex] by the leading term of the divisor [tex]\( 4x \)[/tex]:
[tex]\[
\frac{36x^2}{4x} = 9x
\][/tex]
- Multiply the divisor [tex]\( 4x - 5 \)[/tex] by [tex]\( 9x \)[/tex] and subtract from the current dividend:
[tex]\[
(36x^2 - 25x - 25) - (9x \cdot (4x - 5)) = (36x^2 - 25x - 25) - (36x^2 - 45x)
\][/tex]
- Simplify:
[tex]\[
36x^2 - 25x - 25 - 36x^2 + 45x = 20x - 25
\][/tex]
3. Third Division:
- Divide the leading term of the new dividend [tex]\( 20x \)[/tex] by the leading term of the divisor [tex]\( 4x \)[/tex]:
[tex]\[
\frac{20x}{4x} = 5
\][/tex]
- Multiply the divisor [tex]\( 4x - 5 \)[/tex] by [tex]\( 5 \)[/tex] and subtract from the current dividend:
[tex]\[
(20x - 25) - (5 \cdot (4x - 5)) = (20x - 25) - (20x - 25)
\][/tex]
- Simplify:
[tex]\[
20x - 25 - 20x + 25 = 0
\][/tex]
Thus, after performing the polynomial long division, we get:
- Quotient: [tex]\( 2x^2 + 9x + 5 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
So, the result of the division is:
[tex]\[
8x^3 + 26x^2 - 25x - 25 = (4x - 5) \cdot (2x^2 + 9x + 5) + 0
\][/tex]
In conclusion:
1. The quotient [tex]\( q(x) \)[/tex] is [tex]\( 2x^2 + 9x + 5 \)[/tex].
2. The remainder [tex]\( r(x) \)[/tex] is [tex]\( 0 \)[/tex].
