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Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], you can perform polynomial long division. Here's how you can do it step-by-step:
1. Set up the division: Write down the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the first terms: Take the leading term of the dividend, [tex]\(x^4\)[/tex], and divide it by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3.
\][/tex]
4. Repeat the process:
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = -3x.
\][/tex]
As the subtraction indicates there's no more term that can be divided by [tex]\(x^3 - 3\)[/tex] (i.e., the degree of the resulting polynomial is less than that of the divisor), you reach the end of the division process. In this case, there is no remainder.
Following these steps gives you the quotient of [tex]\(x + 5\)[/tex] with a remainder of 0.
Therefore, the quotient of the division is [tex]\(x + 5\)[/tex].
1. Set up the division: Write down the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the first terms: Take the leading term of the dividend, [tex]\(x^4\)[/tex], and divide it by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3.
\][/tex]
4. Repeat the process:
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 3x - 15) - (5x^3 - 15) = -3x.
\][/tex]
As the subtraction indicates there's no more term that can be divided by [tex]\(x^3 - 3\)[/tex] (i.e., the degree of the resulting polynomial is less than that of the divisor), you reach the end of the division process. In this case, there is no remainder.
Following these steps gives you the quotient of [tex]\(x + 5\)[/tex] with a remainder of 0.
Therefore, the quotient of the division is [tex]\(x + 5\)[/tex].
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