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Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we need to perform polynomial long division. Here's a step-by-step explanation of how this is done:
1. Set Up the Division:
- The dividend (numerator) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor (denominator) is [tex]\(x^3 - 3\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- This gives the first term of the quotient: [tex]\(x^4 \div x^3 = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first quotient term [tex]\(x\)[/tex], which results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend. This yields:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2
\][/tex]
4. Repeat the Process:
- Now take the new polynomial [tex]\(5x^3 + 0x^2\)[/tex] and repeat the division process. Divide the leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex], yielding [tex]\(5\)[/tex].
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this result from the current dividend:
[tex]\[
(5x^3 + 0x^2) - (5x^3 - 15) = 15
\][/tex]
5. Remainder:
- After subtraction, we are left with [tex]\(15\)[/tex], which is a constant and the remainder.
- Since we are only interested in the quotient, and the problem specifies that the quotient is a polynomial, we stop here.
6. Result:
- Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
So, the polynomial that represents the quotient is [tex]\(x + 5\)[/tex].
1. Set Up the Division:
- The dividend (numerator) is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor (denominator) is [tex]\(x^3 - 3\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- This gives the first term of the quotient: [tex]\(x^4 \div x^3 = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first quotient term [tex]\(x\)[/tex], which results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend. This yields:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2
\][/tex]
4. Repeat the Process:
- Now take the new polynomial [tex]\(5x^3 + 0x^2\)[/tex] and repeat the division process. Divide the leading term, [tex]\(5x^3\)[/tex], by [tex]\(x^3\)[/tex], yielding [tex]\(5\)[/tex].
- Multiply the divisor by [tex]\(5\)[/tex] to get [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this result from the current dividend:
[tex]\[
(5x^3 + 0x^2) - (5x^3 - 15) = 15
\][/tex]
5. Remainder:
- After subtraction, we are left with [tex]\(15\)[/tex], which is a constant and the remainder.
- Since we are only interested in the quotient, and the problem specifies that the quotient is a polynomial, we stop here.
6. Result:
- Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
So, the polynomial that represents the quotient is [tex]\(x + 5\)[/tex].
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