Answer :
To find the quotient of the division of two polynomials, [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division.
Here is the step-by-step process:
1. Divide the first term of the dividend by the first term of the divisor:
The first term of the dividend is [tex]\(x^4\)[/tex], and the first term of the divisor is [tex]\(x^3\)[/tex]. Dividing these gives:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply the entire divisor by this result:
Multiply [tex]\(x\)[/tex] by [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this from the original dividend:
Subtract [tex]\((x^4 - 3x)\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Bring down the next term:
After subtraction, the current expression is [tex]\(5x^3 - 15\)[/tex].
5. Repeat the division process with the new polynomial:
Divide the first term ([tex]\(5x^3\)[/tex]) by the first term of the divisor ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
6. Multiply the entire divisor by this new result:
Multiply [tex]\(5\)[/tex] by [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
7. Subtract this from the current polynomial:
Subtract [tex]\((5x^3 - 15)\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
At this point, the remainder is zero, which confirms that our division process is complete, and the quotient is a polynomial [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].
Here is the step-by-step process:
1. Divide the first term of the dividend by the first term of the divisor:
The first term of the dividend is [tex]\(x^4\)[/tex], and the first term of the divisor is [tex]\(x^3\)[/tex]. Dividing these gives:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply the entire divisor by this result:
Multiply [tex]\(x\)[/tex] by [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this from the original dividend:
Subtract [tex]\((x^4 - 3x)\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Bring down the next term:
After subtraction, the current expression is [tex]\(5x^3 - 15\)[/tex].
5. Repeat the division process with the new polynomial:
Divide the first term ([tex]\(5x^3\)[/tex]) by the first term of the divisor ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
6. Multiply the entire divisor by this new result:
Multiply [tex]\(5\)[/tex] by [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
7. Subtract this from the current polynomial:
Subtract [tex]\((5x^3 - 15)\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
At this point, the remainder is zero, which confirms that our division process is complete, and the quotient is a polynomial [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].
