To solve the expression [tex]\(\frac{5x^3 + x - 3}{x - 1}\)[/tex] by performing polynomial division, follow these steps:
1. Set up the division:
We start by dividing the polynomial [tex]\(5x^3 + x - 3\)[/tex] by [tex]\(x - 1\)[/tex].
2. Divide the leading terms:
- Take the leading term of the numerator [tex]\(5x^3\)[/tex] and divide it by the leading term of the denominator [tex]\(x\)[/tex]:
[tex]\[
\frac{5x^3}{x} = 5x^2
\][/tex]
- So, the first term of the quotient is [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(x - 1\)[/tex]:
[tex]\[
5x^2 \cdot (x - 1) = 5x^3 - 5x^2
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(5x^3 + x - 3) - (5x^3 - 5x^2) = 0 + 5x^2 + x - 3
\][/tex]
4. Repeat the process:
- Now divide [tex]\(5x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{5x^2}{x} = 5x
\][/tex]
- The next term of the quotient is [tex]\(5x\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(x - 1\)[/tex]:
[tex]\[
5x \cdot (x - 1) = 5x^2 - 5x
\][/tex]
- Subtract:
[tex]\[
(5x^2 + x - 3) - (5x^2 - 5x) = 0 + 6x - 3
\][/tex]
5. Continue the division:
- Divide [tex]\(6x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{6x}{x} = 6
\][/tex]
- The next term of the quotient is [tex]\(6\)[/tex].
- Multiply [tex]\(6\)[/tex] by [tex]\(x - 1\)[/tex]:
[tex]\[
6 \cdot (x - 1) = 6x - 6
\][/tex]
- Subtract:
[tex]\[
(6x - 3) - (6x - 6) = 0 + 3
\][/tex]
6. Combine the results:
- The quotient we have found so far is [tex]\(5x^2 + 5x + 6\)[/tex], and the remainder is [tex]\(3\)[/tex].
Thus, when [tex]\(\frac{5x^3 + x - 3}{x - 1}\)[/tex] is divided, we get:
[tex]\[
\frac{5x^3 + x - 3}{x - 1} = 5x^2 + 5x + 6 + \frac{3}{x - 1}
\][/tex]
This tells us that the quotient is [tex]\(5x^2 + 5x + 6\)[/tex] and the remainder is [tex]\(3\)[/tex].
