Answer :
To perform the division of the polynomial [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex] by [tex]\(x - 5\)[/tex], we can use polynomial long division. Here's how we can solve it step by step:
1. Set Up the Division:
- Dividend: [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex]
- Divisor: [tex]\(x - 5\)[/tex]
2. Divide the First Term:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{x^4}{x} = x^3
\][/tex]
- This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\(x^3\)[/tex] by the whole divisor [tex]\(x - 5\)[/tex]:
[tex]\[
x^3(x - 5) = x^4 - 5x^3
\][/tex]
- Subtract this from the original dividend:
[tex]\[
(x^4 - x^3 - 19x^2 - 3x - 19) - (x^4 - 5x^3) = 4x^3 - 19x^2 - 3x - 19
\][/tex]
4. Repeat the Process:
- Divide the first term of the new dividend [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{4x^3}{x} = 4x^2
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by the divisor [tex]\(x - 5\)[/tex]:
[tex]\[
4x^2(x - 5) = 4x^3 - 20x^2
\][/tex]
- Subtract from the current dividend:
[tex]\[
(4x^3 - 19x^2 - 3x - 19) - (4x^3 - 20x^2) = x^2 - 3x - 19
\][/tex]
5. Continue the Process:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[
x(x - 5) = x^2 - 5x
\][/tex]
- Subtract:
[tex]\[
(x^2 - 3x - 19) - (x^2 - 5x) = 2x - 19
\][/tex]
6. Final Step:
- Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{x} = 2
\][/tex]
- Multiply [tex]\(2\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[
2(x - 5) = 2x - 10
\][/tex]
- Subtract:
[tex]\[
(2x - 19) - (2x - 10) = -9
\][/tex]
So, the quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
Thus, the result of the division is:
[tex]\[
x^3 + 4x^2 + x + 2 \quad \text{with a remainder of} \quad -9
\][/tex]
1. Set Up the Division:
- Dividend: [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex]
- Divisor: [tex]\(x - 5\)[/tex]
2. Divide the First Term:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{x^4}{x} = x^3
\][/tex]
- This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\(x^3\)[/tex] by the whole divisor [tex]\(x - 5\)[/tex]:
[tex]\[
x^3(x - 5) = x^4 - 5x^3
\][/tex]
- Subtract this from the original dividend:
[tex]\[
(x^4 - x^3 - 19x^2 - 3x - 19) - (x^4 - 5x^3) = 4x^3 - 19x^2 - 3x - 19
\][/tex]
4. Repeat the Process:
- Divide the first term of the new dividend [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{4x^3}{x} = 4x^2
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by the divisor [tex]\(x - 5\)[/tex]:
[tex]\[
4x^2(x - 5) = 4x^3 - 20x^2
\][/tex]
- Subtract from the current dividend:
[tex]\[
(4x^3 - 19x^2 - 3x - 19) - (4x^3 - 20x^2) = x^2 - 3x - 19
\][/tex]
5. Continue the Process:
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[
x(x - 5) = x^2 - 5x
\][/tex]
- Subtract:
[tex]\[
(x^2 - 3x - 19) - (x^2 - 5x) = 2x - 19
\][/tex]
6. Final Step:
- Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{x} = 2
\][/tex]
- Multiply [tex]\(2\)[/tex] by [tex]\(x - 5\)[/tex]:
[tex]\[
2(x - 5) = 2x - 10
\][/tex]
- Subtract:
[tex]\[
(2x - 19) - (2x - 10) = -9
\][/tex]
So, the quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
Thus, the result of the division is:
[tex]\[
x^3 + 4x^2 + x + 2 \quad \text{with a remainder of} \quad -9
\][/tex]
