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Answer :
To divide 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 a step-by-step explanation:
1. Set Up the Division:
Write [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex] inside the division symbol and [tex]\(x - 5\)[/tex] outside.
2. Divide the First Terms:
Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex].
[tex]\(x^4 \div x = x^3\)[/tex].
Write [tex]\(x^3\)[/tex] above the division line.
3. Multiply and Subtract:
Multiply the entire divisor, [tex]\(x - 5\)[/tex], by [tex]\(x^3\)[/tex] to get [tex]\(x^4 - 5x^3\)[/tex].
Subtract this from the original polynomial.
[tex]\((x^4 - x^3 - 19x^2) - (x^4 - 5x^3) = 4x^3 - 19x^2\)[/tex].
4. Repeat the Process:
Bring down the next term, [tex]\(-3x\)[/tex], to get [tex]\(4x^3 - 19x^2 - 3x\)[/tex].
5. Divide the First Terms Again:
Divide [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(4x^2\)[/tex].
Write [tex]\(4x^2\)[/tex] above the division.
6. Multiply and Subtract:
Multiply [tex]\(x - 5\)[/tex] by [tex]\(4x^2\)[/tex] to get [tex]\(4x^3 - 20x^2\)[/tex].
Subtract: [tex]\((4x^3 - 19x^2) - (4x^3 - 20x^2) = x^2\)[/tex].
7. Continue:
Bring down the next term, [tex]\(-3x\)[/tex], to get [tex]\(x^2 - 3x\)[/tex].
8. Divide Again:
Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x\)[/tex].
Multiply [tex]\(x - 5\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^2 - 5x\)[/tex].
Subtract: [tex]\((x^2 - 3x) - (x^2 - 5x) = 2x\)[/tex].
9. Final Steps:
Bring down the last term, [tex]\(-19\)[/tex], to get [tex]\(2x - 19\)[/tex].
10. Divide for the Last Time:
Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2\)[/tex].
Multiply [tex]\(x - 5\)[/tex] by [tex]\(2\)[/tex] to get [tex]\(2x - 10\)[/tex].
Subtract: [tex]\((2x - 19) - (2x - 10) = -9\)[/tex].
11. Result:
The quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
So, when [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex] is divided by [tex]\(x - 5\)[/tex], the quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
1. Set Up the Division:
Write [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex] inside the division symbol and [tex]\(x - 5\)[/tex] outside.
2. Divide the First Terms:
Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex].
[tex]\(x^4 \div x = x^3\)[/tex].
Write [tex]\(x^3\)[/tex] above the division line.
3. Multiply and Subtract:
Multiply the entire divisor, [tex]\(x - 5\)[/tex], by [tex]\(x^3\)[/tex] to get [tex]\(x^4 - 5x^3\)[/tex].
Subtract this from the original polynomial.
[tex]\((x^4 - x^3 - 19x^2) - (x^4 - 5x^3) = 4x^3 - 19x^2\)[/tex].
4. Repeat the Process:
Bring down the next term, [tex]\(-3x\)[/tex], to get [tex]\(4x^3 - 19x^2 - 3x\)[/tex].
5. Divide the First Terms Again:
Divide [tex]\(4x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(4x^2\)[/tex].
Write [tex]\(4x^2\)[/tex] above the division.
6. Multiply and Subtract:
Multiply [tex]\(x - 5\)[/tex] by [tex]\(4x^2\)[/tex] to get [tex]\(4x^3 - 20x^2\)[/tex].
Subtract: [tex]\((4x^3 - 19x^2) - (4x^3 - 20x^2) = x^2\)[/tex].
7. Continue:
Bring down the next term, [tex]\(-3x\)[/tex], to get [tex]\(x^2 - 3x\)[/tex].
8. Divide Again:
Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x\)[/tex].
Multiply [tex]\(x - 5\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^2 - 5x\)[/tex].
Subtract: [tex]\((x^2 - 3x) - (x^2 - 5x) = 2x\)[/tex].
9. Final Steps:
Bring down the last term, [tex]\(-19\)[/tex], to get [tex]\(2x - 19\)[/tex].
10. Divide for the Last Time:
Divide [tex]\(2x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(2\)[/tex].
Multiply [tex]\(x - 5\)[/tex] by [tex]\(2\)[/tex] to get [tex]\(2x - 10\)[/tex].
Subtract: [tex]\((2x - 19) - (2x - 10) = -9\)[/tex].
11. Result:
The quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
So, when [tex]\(x^4 - x^3 - 19x^2 - 3x - 19\)[/tex] is divided by [tex]\(x - 5\)[/tex], the quotient is [tex]\(x^3 + 4x^2 + x + 2\)[/tex] and the remainder is [tex]\(-9\)[/tex].
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