Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we use polynomial division, similar to long division with numbers. Here's how it works step-by-step:
1. Set up the division:
The dividend is [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and the divisor is [tex]\((x^3 - 3)\)[/tex].
2. Divide the first terms:
Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex]:
[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 [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 -15
\][/tex]
4. Repeat the process:
Since we have reduced the degree of the polynomial, compare it again with the divisor:
Divide the first term of the remaining polynomial, [tex]\(5x^3\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by this new term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract again:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
At this point, we have reached a remainder of 0, indicating that the division is complete. The quotient obtained from the division 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:
The dividend is [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and the divisor is [tex]\((x^3 - 3)\)[/tex].
2. Divide the first terms:
Divide the first term of the dividend, [tex]\(x^4\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex]:
[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 [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 -15
\][/tex]
4. Repeat the process:
Since we have reduced the degree of the polynomial, compare it again with the divisor:
Divide the first term of the remaining polynomial, [tex]\(5x^3\)[/tex], by the first term of the divisor, [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by this new term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract again:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
At this point, we have reached a remainder of 0, indicating that the division is complete. The quotient obtained from the division 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].
