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Answer :
To solve this problem, we need to perform polynomial division on the given expressions:
1. Identify the polynomial expressions:
- The numerator is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The denominator is [tex]\( x^3 - 3 \)[/tex].
2. Perform polynomial division:
- We'll divide [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
3. Long division process:
- Step 1: Divide the leading term of the numerator, [tex]\( x^4 \)[/tex], by the leading term of the denominator, [tex]\( x^3 \)[/tex]. This gives us the first term of the quotient: [tex]\( x \)[/tex].
- Step 2: Multiply the entire denominator [tex]\( x^3 - 3 \)[/tex] by this term [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Step 3: Subtract [tex]\( x^4 - 3x \)[/tex] from the numerator [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]. This results in [tex]\( 5x^3 + 0x^2 + 0 - 15 \)[/tex].
- Step 4: Now, divide the new leading term of the result [tex]\( 5x^3 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex] to get the next term in the quotient: [tex]\( +5 \)[/tex].
- Step 5: Multiply the entire denominator [tex]\( x^3 - 3 \)[/tex] by this new term [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Step 6: Subtract [tex]\( 5x^3 - 15 \)[/tex] from the current polynomial [tex]\( 5x^3 + 0x^2 + 0 - 15 \)[/tex]. This yields a remainder of 0.
Since we have reached a remainder of 0, the division process is complete. The quotient is a polynomial with no remainder.
4. Final Result:
- The quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\( x + 5 \)[/tex].
Therefore, the correct answer is: [tex]\( x + 5 \)[/tex].
1. Identify the polynomial expressions:
- The numerator is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The denominator is [tex]\( x^3 - 3 \)[/tex].
2. Perform polynomial division:
- We'll divide [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
3. Long division process:
- Step 1: Divide the leading term of the numerator, [tex]\( x^4 \)[/tex], by the leading term of the denominator, [tex]\( x^3 \)[/tex]. This gives us the first term of the quotient: [tex]\( x \)[/tex].
- Step 2: Multiply the entire denominator [tex]\( x^3 - 3 \)[/tex] by this term [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Step 3: Subtract [tex]\( x^4 - 3x \)[/tex] from the numerator [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]. This results in [tex]\( 5x^3 + 0x^2 + 0 - 15 \)[/tex].
- Step 4: Now, divide the new leading term of the result [tex]\( 5x^3 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex] to get the next term in the quotient: [tex]\( +5 \)[/tex].
- Step 5: Multiply the entire denominator [tex]\( x^3 - 3 \)[/tex] by this new term [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Step 6: Subtract [tex]\( 5x^3 - 15 \)[/tex] from the current polynomial [tex]\( 5x^3 + 0x^2 + 0 - 15 \)[/tex]. This yields a remainder of 0.
Since we have reached a remainder of 0, the division process is complete. The quotient is a polynomial with no remainder.
4. Final Result:
- The quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is [tex]\( x + 5 \)[/tex].
Therefore, the correct answer is: [tex]\( x + 5 \)[/tex].
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