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Answer :
To divide the polynomial [tex]\(20x^3 + 19x^2 - 16x - 13\)[/tex] by [tex]\(5x + 6\)[/tex], we will use polynomial long division. Here's a step-by-step procedure:
1. Divide the first term: Look at the leading term of the dividend, [tex]\(20x^3\)[/tex]. Divide this by the leading term of the divisor, [tex]\(5x\)[/tex], giving [tex]\(4x^2\)[/tex].
2. Multiply and subtract: Multiply [tex]\(4x^2\)[/tex] by the divisor [tex]\(5x + 6\)[/tex], which results in [tex]\(20x^3 + 24x^2\)[/tex]. Subtract this from the original dividend:
[tex]\[
(20x^3 + 19x^2 - 16x - 13) - (20x^3 + 24x^2) = -5x^2 - 16x - 13
\][/tex]
3. Repeat the process: Now divide [tex]\(-5x^2\)[/tex] by [tex]\(5x\)[/tex], resulting in [tex]\(-x\)[/tex]. Multiply [tex]\(-x\)[/tex] by the divisor [tex]\(5x + 6\)[/tex] to get [tex]\(-5x^2 - 6x\)[/tex]. Subtract this from [tex]\(-5x^2 - 16x - 13\)[/tex]:
[tex]\[
(-5x^2 - 16x - 13) - (-5x^2 - 6x) = -10x - 13
\][/tex]
4. Repeat once more: Divide [tex]\(-10x\)[/tex] by [tex]\(5x\)[/tex], resulting in [tex]\(-2\)[/tex]. Multiply [tex]\(-2\)[/tex] by the divisor [tex]\(5x + 6\)[/tex] to get [tex]\(-10x - 12\)[/tex]. Subtract this from [tex]\(-10x - 13\)[/tex]:
[tex]\[
(-10x - 13) - (-10x - 12) = -1
\][/tex]
5. Conclusion: The division yields a quotient of [tex]\(4x^2 - x - 2\)[/tex] and a remainder of [tex]\(-1\)[/tex].
So, the quotient is [tex]\(4x^2 - x - 2\)[/tex] and the remainder is [tex]\(-1\)[/tex].
1. Divide the first term: Look at the leading term of the dividend, [tex]\(20x^3\)[/tex]. Divide this by the leading term of the divisor, [tex]\(5x\)[/tex], giving [tex]\(4x^2\)[/tex].
2. Multiply and subtract: Multiply [tex]\(4x^2\)[/tex] by the divisor [tex]\(5x + 6\)[/tex], which results in [tex]\(20x^3 + 24x^2\)[/tex]. Subtract this from the original dividend:
[tex]\[
(20x^3 + 19x^2 - 16x - 13) - (20x^3 + 24x^2) = -5x^2 - 16x - 13
\][/tex]
3. Repeat the process: Now divide [tex]\(-5x^2\)[/tex] by [tex]\(5x\)[/tex], resulting in [tex]\(-x\)[/tex]. Multiply [tex]\(-x\)[/tex] by the divisor [tex]\(5x + 6\)[/tex] to get [tex]\(-5x^2 - 6x\)[/tex]. Subtract this from [tex]\(-5x^2 - 16x - 13\)[/tex]:
[tex]\[
(-5x^2 - 16x - 13) - (-5x^2 - 6x) = -10x - 13
\][/tex]
4. Repeat once more: Divide [tex]\(-10x\)[/tex] by [tex]\(5x\)[/tex], resulting in [tex]\(-2\)[/tex]. Multiply [tex]\(-2\)[/tex] by the divisor [tex]\(5x + 6\)[/tex] to get [tex]\(-10x - 12\)[/tex]. Subtract this from [tex]\(-10x - 13\)[/tex]:
[tex]\[
(-10x - 13) - (-10x - 12) = -1
\][/tex]
5. Conclusion: The division yields a quotient of [tex]\(4x^2 - x - 2\)[/tex] and a remainder of [tex]\(-1\)[/tex].
So, the quotient is [tex]\(4x^2 - x - 2\)[/tex] and the remainder is [tex]\(-1\)[/tex].
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