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Answer :
To divide the polynomial [tex]\(4x^4 - 23x^3 + 16x^2 - 8x + 15\)[/tex] by [tex]\(x - 5\)[/tex], we can use polynomial long division. Here’s how the process works step by step:
1. Set Up the Division: Write the dividend [tex]\(4x^4 - 23x^3 + 16x^2 - 8x + 15\)[/tex] under the division bracket and the divisor [tex]\(x - 5\)[/tex] outside.
2. First Division:
- Divide the first term of the dividend [tex]\(4x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex], which gives [tex]\(4x^3\)[/tex].
- Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex], resulting in [tex]\(4x^4 - 20x^3\)[/tex].
- Subtract [tex]\(4x^4 - 20x^3\)[/tex] from the original polynomial, giving us the new dividend of [tex]\(-3x^3 + 16x^2 - 8x + 15\)[/tex].
3. Second Division:
- Divide the first term of the new dividend [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
- Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 5\)[/tex], resulting in [tex]\(-3x^3 + 15x^2\)[/tex].
- Subtract [tex]\(-3x^3 + 15x^2\)[/tex] from the current dividend, resulting in [tex]\(x^2 - 8x + 15\)[/tex].
4. Third Division:
- Divide the first term of the new dividend [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex], resulting in [tex]\(x^2 - 5x\)[/tex].
- Subtract [tex]\(x^2 - 5x\)[/tex] from the current dividend, which results in [tex]\(-3x + 15\)[/tex].
5. Fourth Division:
- Divide the first term of the new dividend [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(x - 5\)[/tex], which results in [tex]\(-3x + 15\)[/tex].
- Subtract [tex]\(-3x + 15\)[/tex] from the current dividend, leaving a remainder of 0.
Since the remainder is 0, the division is exact.
The quotient of the division is:
[tex]\[ 4x^3 - 3x^2 + x - 3 \][/tex]
There is no remainder. Thus, the solution to the division is the polynomial [tex]\(4x^3 - 3x^2 + x - 3\)[/tex].
1. Set Up the Division: Write the dividend [tex]\(4x^4 - 23x^3 + 16x^2 - 8x + 15\)[/tex] under the division bracket and the divisor [tex]\(x - 5\)[/tex] outside.
2. First Division:
- Divide the first term of the dividend [tex]\(4x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex], which gives [tex]\(4x^3\)[/tex].
- Multiply [tex]\(4x^3\)[/tex] by the entire divisor [tex]\(x - 5\)[/tex], resulting in [tex]\(4x^4 - 20x^3\)[/tex].
- Subtract [tex]\(4x^4 - 20x^3\)[/tex] from the original polynomial, giving us the new dividend of [tex]\(-3x^3 + 16x^2 - 8x + 15\)[/tex].
3. Second Division:
- Divide the first term of the new dividend [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
- Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 5\)[/tex], resulting in [tex]\(-3x^3 + 15x^2\)[/tex].
- Subtract [tex]\(-3x^3 + 15x^2\)[/tex] from the current dividend, resulting in [tex]\(x^2 - 8x + 15\)[/tex].
4. Third Division:
- Divide the first term of the new dividend [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by [tex]\(x - 5\)[/tex], resulting in [tex]\(x^2 - 5x\)[/tex].
- Subtract [tex]\(x^2 - 5x\)[/tex] from the current dividend, which results in [tex]\(-3x + 15\)[/tex].
5. Fourth Division:
- Divide the first term of the new dividend [tex]\(-3x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(x - 5\)[/tex], which results in [tex]\(-3x + 15\)[/tex].
- Subtract [tex]\(-3x + 15\)[/tex] from the current dividend, leaving a remainder of 0.
Since the remainder is 0, the division is exact.
The quotient of the division is:
[tex]\[ 4x^3 - 3x^2 + x - 3 \][/tex]
There is no remainder. Thus, the solution to the division is the polynomial [tex]\(4x^3 - 3x^2 + x - 3\)[/tex].
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