Answer :
To solve the problem of finding the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], you can follow these steps:
1. Set up the Division: You want to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Perform Polynomial Division:
- Divide the leading term of the dividend, which is [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get the first term of the quotient: [tex]\(x^4 \div x^3 = x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the first term of the quotient, [tex]\(x\)[/tex], to get: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15 + 3x = 5x^3 + 3x - 15
\][/tex]
- Repeat the process with this new polynomial:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get the next term of the quotient: [tex]\(5x^3 \div x^3 = 5\)[/tex].
- Multiply the entire divisor by this term: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract again:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x - 15 + 15 = 3x
\][/tex]
- At this point, the degree of the remainder [tex]\(3x\)[/tex] is less than the degree of the divisor, so the division process stops here.
3. Write Out the Quotient:
- The terms you accumulated during division form the quotient, which is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
1. Set up the Division: You want to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Perform Polynomial Division:
- Divide the leading term of the dividend, which is [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get the first term of the quotient: [tex]\(x^4 \div x^3 = x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the first term of the quotient, [tex]\(x\)[/tex], to get: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15 + 3x = 5x^3 + 3x - 15
\][/tex]
- Repeat the process with this new polynomial:
- Divide the new leading term, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], to get the next term of the quotient: [tex]\(5x^3 \div x^3 = 5\)[/tex].
- Multiply the entire divisor by this term: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract again:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x - 15 + 15 = 3x
\][/tex]
- At this point, the degree of the remainder [tex]\(3x\)[/tex] is less than the degree of the divisor, so the division process stops here.
3. Write Out the Quotient:
- The terms you accumulated during division form the quotient, which is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
