Sure, let's go through the polynomial long division step-by-step to divide [tex]\( -30 x^6 + 48 x^5 + 50 x - 80 \)[/tex] by [tex]\( 5 x - 8 \)[/tex].
### Step-by-Step Polynomial Division:
1. Setup the Division:
Write the dividend ([tex]\( -30 x^6 + 48 x^5 + 50 x - 80 \)[/tex]) under the division bar, and the divisor ([tex]\( 5 x - 8 \)[/tex]) outside the division bar.
2. Determine the First Term of the Quotient:
Divide the leading term of the dividend ([tex]\( -30 x^6 \)[/tex]) by the leading term of the divisor ([tex]\( 5 x \)[/tex]).
[tex]\[
\frac{-30 x^6}{5 x} = -6 x^5
\][/tex]
So, the first term of the quotient is [tex]\( -6 x^5 \)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor [tex]\( 5 x - 8 \)[/tex] by the first term of the quotient [tex]\( -6 x^5 \)[/tex]:
[tex]\[
-6 x^5 \cdot (5 x - 8) = -30 x^6 + 48 x^5
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(-30 x^6 + 48 x^5 + 50 x - 80) - (-30 x^6 + 48 x^5) = 50 x - 80
\][/tex]
Now, the new dividend is [tex]\( 50 x - 80 \)[/tex].
4. Determine the Next Term of the Quotient:
Divide the leading term of the new dividend ([tex]\( 50 x \)[/tex]) by the leading term of the divisor ([tex]\( 5 x \)[/tex]):
[tex]\[
\frac{50 x}{5 x} = 10
\][/tex]
So, the next term of the quotient is [tex]\( 10 \)[/tex].
5. Multiply and Subtract Again:
Multiply the entire divisor [tex]\( 5 x - 8 \)[/tex] by the next term of the quotient [tex]\( 10 \)[/tex]:
[tex]\[
10 \cdot (5 x - 8) = 50 x - 80
\][/tex]
Subtract this result from the new dividend:
[tex]\[
(50 x - 80) - (50 x - 80) = 0
\][/tex]
Now, we have a remainder of [tex]\( 0 \)[/tex].
### Final Quotient and Remainder:
The quotient of the division is [tex]\( -6 x^5 + 10 \)[/tex], and the remainder is [tex]\( 0 \)[/tex].
Hence, the division of [tex]\( -30 x^6 + 48 x^5 + 50 x - 80 \)[/tex] by [tex]\( 5 x - 8 \)[/tex] results in:
[tex]\[
\boxed{-6x^5 + 10}
\][/tex]
with a remainder of [tex]\( 0 \)[/tex].
