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Answer :
Let's find the quotient of the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex].
We'll use polynomial long division:
1. 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].
2. Multiply the entire divisor by this term:
[tex]\[
x \times (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 0 - (-3x) \Rightarrow 5x^3 - 15
\][/tex]
4. Repeat the process with the new polynomial:
- [tex]\( \frac{5x^3}{x^3} = 5 \)[/tex]
- So, our quotient becomes [tex]\( x + 5 \)[/tex].
5. Multiply the entire divisor by this new term:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we're left with a remainder of zero, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is simply:
[tex]\( x + 5 \)[/tex].
So, the answer is: [tex]\( x + 5 \)[/tex].
We'll use polynomial long division:
1. 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].
2. Multiply the entire divisor by this term:
[tex]\[
x \times (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 0 - (-3x) \Rightarrow 5x^3 - 15
\][/tex]
4. Repeat the process with the new polynomial:
- [tex]\( \frac{5x^3}{x^3} = 5 \)[/tex]
- So, our quotient becomes [tex]\( x + 5 \)[/tex].
5. Multiply the entire divisor by this new term:
[tex]\[
5 \times (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we're left with a remainder of zero, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is simply:
[tex]\( x + 5 \)[/tex].
So, the answer is: [tex]\( x + 5 \)[/tex].
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