To perform the division [tex]\((5x^4 - x^3 + 45x - 11) \div (5x - 1)\)[/tex], we will follow the process of polynomial long division step by step.
1. Setup the division:
[tex]\[
\frac{5x^4 - x^3 + 45x - 11}{5x - 1}
\][/tex]
2. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{5x^4}{5x} = x^3
\][/tex]
3. Multiply the entire divisor by the result obtained in the previous step:
[tex]\[
x^3 \cdot (5x - 1) = 5x^4 - x^3
\][/tex]
4. Subtract this result from the original numerator:
[tex]\[
(5x^4 - x^3 + 45x - 11) - (5x^4 - x^3) = (5x^4 - x^3 + 45x - 11) - 5x^4 + x^3 = 0 + 0 + 45x - 11 = 45x - 11
\][/tex]
5. We now repeat the same process with the new polynomial [tex]\(45x - 11\)[/tex]. Divide the leading term [tex]\(45x\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[
\frac{45x}{5x} = 9
\][/tex]
6. Multiply the entire divisor by 9:
[tex]\[
9 \cdot (5x - 1) = 45x - 9
\][/tex]
7. Subtract this result from the current polynomial:
[tex]\[
(45x - 11) - (45x - 9) = (45x - 11) - 45x + 9 = 0 - 2 = -2
\][/tex]
8. Finally, because the remainder [tex]\(-2\)[/tex] is of lower degree than the divisor [tex]\(5x - 1\)[/tex], our process is complete. So, we have:
Quotient: [tex]\(x^3 + 9\)[/tex]
Remainder: [tex]\(-2\)[/tex]
Thus, the division of [tex]\(5x^4 - x^3 + 45x - 11\)[/tex] by [tex]\(5x - 1\)[/tex] gives a quotient of [tex]\(x^3 + 9\)[/tex] and a remainder of [tex]\(-2\)[/tex].
