Answer :
Certainly! Let's solve the problem of finding the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex].
### Step-by-Step Solution:
1. Arrange the Polynomials:
- Numerator (dividend): [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Denominator (divisor): [tex]\(x^3 - 3\)[/tex]
2. Perform Division:
- We want to divide the highest degree term of the numerator, [tex]\(x^4\)[/tex], by the highest degree term of the denominator, [tex]\(x^3\)[/tex].
3. First Term of the Quotient:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
x^4 \div x^3 = x
\][/tex]
- The first term of the quotient is [tex]\(x\)[/tex].
4. Multiply and Subtract:
- Multiply the entire divisor by this term of the quotient:
[tex]\[
x \times (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
5. Repeat the Process:
- Now, take the new polynomial [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
5x^3 \div x^3 = 5
\][/tex]
- Add [tex]\(5\)[/tex] to the quotient.
6. Multiply and Subtract Again:
- Multiply the divisor by this new term:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion:
- The remainder is zero, which indicates that [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divides perfectly by [tex]\((x^3 - 3)\)[/tex].
8. Final Answer:
- The quotient 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 the polynomial [tex]\(x + 5\)[/tex].
### Step-by-Step Solution:
1. Arrange the Polynomials:
- Numerator (dividend): [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Denominator (divisor): [tex]\(x^3 - 3\)[/tex]
2. Perform Division:
- We want to divide the highest degree term of the numerator, [tex]\(x^4\)[/tex], by the highest degree term of the denominator, [tex]\(x^3\)[/tex].
3. First Term of the Quotient:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
x^4 \div x^3 = x
\][/tex]
- The first term of the quotient is [tex]\(x\)[/tex].
4. Multiply and Subtract:
- Multiply the entire divisor by this term of the quotient:
[tex]\[
x \times (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
5. Repeat the Process:
- Now, take the new polynomial [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
5x^3 \div x^3 = 5
\][/tex]
- Add [tex]\(5\)[/tex] to the quotient.
6. Multiply and Subtract Again:
- Multiply the divisor by this new term:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion:
- The remainder is zero, which indicates that [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divides perfectly by [tex]\((x^3 - 3)\)[/tex].
8. Final Answer:
- The quotient 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 the polynomial [tex]\(x + 5\)[/tex].
