Answer :
To solve the problem of finding the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], follow the polynomial division steps:
1. Set up the Division:
We're dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the Leading Terms:
Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [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 [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
Which simplifies to:
[tex]\[
5x^3 - 15
\][/tex]
4. Repeat the Process:
Notice the leading term [tex]\(5x^3\)[/tex]. Divide by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the second term of the quotient is [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
Multiply [tex]\(5\)[/tex] by the divisor:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract from the previous result:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclude the Division:
Since the remainder is [tex]\(0\)[/tex], we can conclude that the division is exact, and 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 [tex]\(x + 5\)[/tex].
1. Set up the Division:
We're dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the Leading Terms:
Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [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 [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
Which simplifies to:
[tex]\[
5x^3 - 15
\][/tex]
4. Repeat the Process:
Notice the leading term [tex]\(5x^3\)[/tex]. Divide by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the second term of the quotient is [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
Multiply [tex]\(5\)[/tex] by the divisor:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract from the previous result:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclude the Division:
Since the remainder is [tex]\(0\)[/tex], we can conclude that the division is exact, and 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 [tex]\(x + 5\)[/tex].
