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Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we'll perform polynomial long division. Here’s how it's done step-by-step:
1. Set up the division: Write the dividend inside the division symbol and the divisor outside, like you would with numbers.
Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the leading terms: Divide the first term of the dividend by the first term of the divisor.
[tex]\(x^4 \div x^3 = x\)[/tex]
3. Multiply and subtract: Multiply the entire divisor by this result ([tex]\(x\)[/tex]) and subtract from the dividend.
Multiply: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex]
Subtract:
[tex]\[
\begin{array}{c}
\phantom{-}(x^4 + 5x^3 - 3x - 15) \\
-(x^4 - 3x) \\
\hline
\phantom{-}5x^3
\end{array}
\][/tex]
[tex]\(5x^3 - 3x\)[/tex]
4. Repeat the process: Again, divide the leading term of this new polynomial ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]).
[tex]\(5x^3 \div x^3 = 5\)[/tex]
5. Multiply and subtract: Multiply the entire divisor by this result ([tex]\(5\)[/tex]) and subtract from the polynomial you currently have.
Multiply: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex]
Subtract:
[tex]\[
\begin{array}{c}
\phantom{-}(5x^3 - 3x - 15) \\
-(5x^3 - 15) \\
\hline
\phantom{-} -3x - 0
\end{array}
\][/tex]
The remainder is [tex]\(-3x - 0 = -3x\)[/tex].
Since the degree of the remainder ([tex]\(-3x\)[/tex]) is lower than the degree of the divisor ([tex]\(x^3 - 3\)[/tex]), we stop here.
So, the quotient is:
[tex]\[ x + 5 \][/tex]
Therefore, the correct answer is [tex]\(x + 5\)[/tex].
1. Set up the division: Write the dividend inside the division symbol and the divisor outside, like you would with numbers.
Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the leading terms: Divide the first term of the dividend by the first term of the divisor.
[tex]\(x^4 \div x^3 = x\)[/tex]
3. Multiply and subtract: Multiply the entire divisor by this result ([tex]\(x\)[/tex]) and subtract from the dividend.
Multiply: [tex]\((x)(x^3 - 3) = x^4 - 3x\)[/tex]
Subtract:
[tex]\[
\begin{array}{c}
\phantom{-}(x^4 + 5x^3 - 3x - 15) \\
-(x^4 - 3x) \\
\hline
\phantom{-}5x^3
\end{array}
\][/tex]
[tex]\(5x^3 - 3x\)[/tex]
4. Repeat the process: Again, divide the leading term of this new polynomial ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]).
[tex]\(5x^3 \div x^3 = 5\)[/tex]
5. Multiply and subtract: Multiply the entire divisor by this result ([tex]\(5\)[/tex]) and subtract from the polynomial you currently have.
Multiply: [tex]\((5)(x^3 - 3) = 5x^3 - 15\)[/tex]
Subtract:
[tex]\[
\begin{array}{c}
\phantom{-}(5x^3 - 3x - 15) \\
-(5x^3 - 15) \\
\hline
\phantom{-} -3x - 0
\end{array}
\][/tex]
The remainder is [tex]\(-3x - 0 = -3x\)[/tex].
Since the degree of the remainder ([tex]\(-3x\)[/tex]) is lower than the degree of the divisor ([tex]\(x^3 - 3\)[/tex]), we stop here.
So, the quotient is:
[tex]\[ x + 5 \][/tex]
Therefore, the correct answer is [tex]\(x + 5\)[/tex].
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