Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division. Here's a step-by-step guide:
1. Set Up the Division: Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Division Step 1:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply the entire divisor by this result: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
3. Division Step 2:
- Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the entire divisor by this result: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the new dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
At this stage, the polynomial we are left with is of degree lower than the divisor [tex]\(x^3 - 3\)[/tex], so no further division is possible. The terms we found, [tex]\(x + 5\)[/tex], make up the quotient of the division.
Therefore, the quotient of the division is [tex]\((x + 5)\)[/tex].
1. Set Up the Division: Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Division Step 1:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply the entire divisor by this result: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
3. Division Step 2:
- Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the entire divisor by this result: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the new dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
At this stage, the polynomial we are left with is of degree lower than the divisor [tex]\(x^3 - 3\)[/tex], so no further division is possible. The terms we found, [tex]\(x + 5\)[/tex], make up the quotient of the division.
Therefore, the quotient of the division is [tex]\((x + 5)\)[/tex].
