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Answer :
To find the quotient and remainder when dividing [tex]\(4x^4 - 19x^3 - 22x - 15\)[/tex] by [tex]\(x - 5\)[/tex], we can use polynomial long division. Here's a step-by-step guide:
Step 1: Set up the division.
Write the dividend [tex]\(4x^4 - 19x^3 + 0x^2 - 22x - 15\)[/tex] and the divisor [tex]\(x - 5\)[/tex].
Step 2: Divide the first term of the dividend by the first term of the divisor.
- [tex]\(\frac{4x^4}{x} = 4x^3\)[/tex].
Write [tex]\(4x^3\)[/tex] as the first term of the quotient.
Step 3: Multiply and subtract.
- Multiply [tex]\(4x^3\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(4x^4 - 20x^3\)[/tex].
- Subtract [tex]\(4x^4 - 20x^3\)[/tex] from [tex]\(4x^4 - 19x^3 + 0x^2 - 22x - 15\)[/tex], which gives you [tex]\(x^3 + 0x^2 - 22x - 15\)[/tex].
Step 4: Repeat the process with the new polynomial.
- Divide the first term of the new polynomial [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{x^3}{x} = x^2\)[/tex].
Write [tex]\(x^2\)[/tex] as the next term of the quotient.
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(x^3 - 5x^2\)[/tex].
- Subtract [tex]\(x^3 - 5x^2\)[/tex] from [tex]\(x^3 + 0x^2 - 22x - 15\)[/tex], which gives you [tex]\(5x^2 - 22x - 15\)[/tex].
Step 5: Continue the process.
- Divide [tex]\(5x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{5x^2}{x} = 5x\)[/tex].
Write [tex]\(5x\)[/tex] as the next term of the quotient.
- Multiply [tex]\(5x\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(5x^2 - 25x\)[/tex].
- Subtract [tex]\(5x^2 - 25x\)[/tex] from [tex]\(5x^2 - 22x - 15\)[/tex], which gives you [tex]\(3x - 15\)[/tex].
Step 6: One more division step.
- Divide [tex]\(3x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{3x}{x} = 3\)[/tex].
Write [tex]\(3\)[/tex] as the next term of the quotient.
- Multiply [tex]\(3\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(3x - 15\)[/tex].
- Subtract [tex]\(3x - 15\)[/tex] from [tex]\(3x - 15\)[/tex], which gives you [tex]\(0\)[/tex].
The remainder is [tex]\(0\)[/tex].
Final Answer:
The quotient is [tex]\(4x^3 + x^2 + 5x + 3\)[/tex].
The remainder is [tex]\(0\)[/tex].
Since the remainder is [tex]\(0\)[/tex], [tex]\((x - 5)\)[/tex] is indeed a factor of [tex]\(4x^4 - 19x^3 - 22x - 15\)[/tex], so the statement is True.
Step 1: Set up the division.
Write the dividend [tex]\(4x^4 - 19x^3 + 0x^2 - 22x - 15\)[/tex] and the divisor [tex]\(x - 5\)[/tex].
Step 2: Divide the first term of the dividend by the first term of the divisor.
- [tex]\(\frac{4x^4}{x} = 4x^3\)[/tex].
Write [tex]\(4x^3\)[/tex] as the first term of the quotient.
Step 3: Multiply and subtract.
- Multiply [tex]\(4x^3\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(4x^4 - 20x^3\)[/tex].
- Subtract [tex]\(4x^4 - 20x^3\)[/tex] from [tex]\(4x^4 - 19x^3 + 0x^2 - 22x - 15\)[/tex], which gives you [tex]\(x^3 + 0x^2 - 22x - 15\)[/tex].
Step 4: Repeat the process with the new polynomial.
- Divide the first term of the new polynomial [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{x^3}{x} = x^2\)[/tex].
Write [tex]\(x^2\)[/tex] as the next term of the quotient.
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(x^3 - 5x^2\)[/tex].
- Subtract [tex]\(x^3 - 5x^2\)[/tex] from [tex]\(x^3 + 0x^2 - 22x - 15\)[/tex], which gives you [tex]\(5x^2 - 22x - 15\)[/tex].
Step 5: Continue the process.
- Divide [tex]\(5x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{5x^2}{x} = 5x\)[/tex].
Write [tex]\(5x\)[/tex] as the next term of the quotient.
- Multiply [tex]\(5x\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(5x^2 - 25x\)[/tex].
- Subtract [tex]\(5x^2 - 25x\)[/tex] from [tex]\(5x^2 - 22x - 15\)[/tex], which gives you [tex]\(3x - 15\)[/tex].
Step 6: One more division step.
- Divide [tex]\(3x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\(\frac{3x}{x} = 3\)[/tex].
Write [tex]\(3\)[/tex] as the next term of the quotient.
- Multiply [tex]\(3\)[/tex] by [tex]\(x - 5\)[/tex] to get [tex]\(3x - 15\)[/tex].
- Subtract [tex]\(3x - 15\)[/tex] from [tex]\(3x - 15\)[/tex], which gives you [tex]\(0\)[/tex].
The remainder is [tex]\(0\)[/tex].
Final Answer:
The quotient is [tex]\(4x^3 + x^2 + 5x + 3\)[/tex].
The remainder is [tex]\(0\)[/tex].
Since the remainder is [tex]\(0\)[/tex], [tex]\((x - 5)\)[/tex] is indeed a factor of [tex]\(4x^4 - 19x^3 - 22x - 15\)[/tex], so the statement is True.
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