Answer :
To find the quotient of the division of two polynomials, [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], we will perform polynomial long division.
### Step-by-Step Solution:
1. Setup the Division:
Place the dividend, [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], under the division symbol, and the divisor, [tex]\(x^3 - 3\)[/tex], outside.
2. Divide the Leading Terms:
Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor by this term [tex]\(x\)[/tex] and subtract the result from the dividend.
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
Subtract:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the Process:
Now treat [tex]\(5x^3 + 0x^2 - 0x - 15\)[/tex] as the new dividend. Divide the leading term by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, add [tex]\(5\)[/tex] to the quotient.
5. Multiply and Subtract Again:
Multiply the entire divisor by this term [tex]\(5\)[/tex] and subtract.
[tex]\[
(x^3 - 3) \times 5 = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5x^3 - 15) = 0x^2 + 0x + 0
\][/tex]
6. Result:
Since the remainder is zero, the division is exact. The quotient polynomial is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
The answer is:
- [tex]\(x + 5\)[/tex]
### Step-by-Step Solution:
1. Setup the Division:
Place the dividend, [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], under the division symbol, and the divisor, [tex]\(x^3 - 3\)[/tex], outside.
2. Divide the Leading Terms:
Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
3. Multiply and Subtract:
Multiply the entire divisor by this term [tex]\(x\)[/tex] and subtract the result from the dividend.
[tex]\[
(x^3 - 3) \times x = x^4 - 3x
\][/tex]
Subtract:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the Process:
Now treat [tex]\(5x^3 + 0x^2 - 0x - 15\)[/tex] as the new dividend. Divide the leading term by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, add [tex]\(5\)[/tex] to the quotient.
5. Multiply and Subtract Again:
Multiply the entire divisor by this term [tex]\(5\)[/tex] and subtract.
[tex]\[
(x^3 - 3) \times 5 = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5x^3 - 15) = 0x^2 + 0x + 0
\][/tex]
6. Result:
Since the remainder is zero, the division is exact. The quotient polynomial is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
The answer is:
- [tex]\(x + 5\)[/tex]
