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Answer :
To find the quotient of the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex], we can perform polynomial long division.
### Step-by-Step Solution:
1. Set Up the Division:
You want to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex]. Align the terms in decreasing order of power.
2. Divide the Leading Terms:
Look at the highest degree terms in the numerator and the denominator. Divide the leading term of the numerator [tex]\( x^4 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Place [tex]\( x \)[/tex] above the division line.
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 numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15 = 5x^3 - 15
\][/tex]
4. Repeat the Process:
Now, divide the new leading term of the result [tex]\( 5x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Place [tex]\( 5 \)[/tex] next to [tex]\( x \)[/tex] in the quotient.
Multiply [tex]\( 5 \)[/tex] by the entire divisor:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The remainder is zero, indicating that the division is exact.
### Conclusion:
The quotient, after performing the polynomial division of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex], is [tex]\( x + 5 \)[/tex] with no remainder. Thus, the answer is:
[tex]\[
x + 5
\][/tex]
### Step-by-Step Solution:
1. Set Up the Division:
You want to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex]. Align the terms in decreasing order of power.
2. Divide the Leading Terms:
Look at the highest degree terms in the numerator and the denominator. Divide the leading term of the numerator [tex]\( x^4 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Place [tex]\( x \)[/tex] above the division line.
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 numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15 = 5x^3 - 15
\][/tex]
4. Repeat the Process:
Now, divide the new leading term of the result [tex]\( 5x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Place [tex]\( 5 \)[/tex] next to [tex]\( x \)[/tex] in the quotient.
Multiply [tex]\( 5 \)[/tex] by the entire divisor:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
The remainder is zero, indicating that the division is exact.
### Conclusion:
The quotient, after performing the polynomial division of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex], is [tex]\( x + 5 \)[/tex] with no remainder. Thus, the answer is:
[tex]\[
x + 5
\][/tex]
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