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Answer :
To solve the problem, we need to find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by the polynomial [tex]\(x^3 - 3\)[/tex]. This can be accomplished through polynomial division. Let's work through the steps:
1. Set up the division:
We are dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex]. Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division symbol and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the leading terms:
Look at the leading term of the dividend [tex]\(x^4\)[/tex] and the leading term of the divisor [tex]\(x^3\)[/tex]. Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the process:
Now, take the new polynomial [tex]\(5x^3 + 3x - 15\)[/tex] as the dividend. Divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex] of the divisor, which gives [tex]\(5\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex]. Subtract this from [tex]\(5x^3 + 3x - 15\)[/tex]:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
6. Finish up:
The remainder is [tex]\(3x\)[/tex], but since the remainder's degree is less than the divisor's degree, the division process stops here.
The polynomial that remains 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].
In conclusion, the answer is: [tex]\(x + 5\)[/tex].
1. Set up the division:
We are dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex]. Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division symbol and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the leading terms:
Look at the leading term of the dividend [tex]\(x^4\)[/tex] and the leading term of the divisor [tex]\(x^3\)[/tex]. Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the process:
Now, take the new polynomial [tex]\(5x^3 + 3x - 15\)[/tex] as the dividend. Divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex] of the divisor, which gives [tex]\(5\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex]. Subtract this from [tex]\(5x^3 + 3x - 15\)[/tex]:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
6. Finish up:
The remainder is [tex]\(3x\)[/tex], but since the remainder's degree is less than the divisor's degree, the division process stops here.
The polynomial that remains 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].
In conclusion, the answer is: [tex]\(x + 5\)[/tex].
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