Answer :
To find the quotient of [tex]\( \left(x^4 + 5x^3 - 3x - 15\right) \)[/tex] divided by [tex]\( \left(x^3 - 3\right) \)[/tex], we can perform polynomial long division. Here is a step-by-step explanation:
1. Setup the Division:
- The dividend (numerator) is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The divisor (denominator) is [tex]\( x^3 - 3 \)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex].
- [tex]\( x^4 \div x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which gives [tex]\( x \times (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which is [tex]\( 5 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by the divisor [tex]\( x^3 - 3 \)[/tex], giving [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
5. Write the Quotient:
- Now, our division has ended because the degree of the remainder [tex]\( 3x \)[/tex] is less than the degree of the divisor [tex]\( x^3 - 3 \)[/tex].
- The quotient of the division is [tex]\( x + 5 \)[/tex].
Therefore, the quotient of [tex]\( \left(x^4 + 5x^3 - 3x - 15\right) \div \left(x^3 - 3\right) \)[/tex] is [tex]\( x + 5 \)[/tex], with a remainder of 0, meaning the division is exact.
1. Setup the Division:
- The dividend (numerator) is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The divisor (denominator) is [tex]\( x^3 - 3 \)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex].
- [tex]\( x^4 \div x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which gives [tex]\( x \times (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which is [tex]\( 5 \)[/tex].
- Multiply [tex]\( 5 \)[/tex] by the divisor [tex]\( x^3 - 3 \)[/tex], giving [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
5. Write the Quotient:
- Now, our division has ended because the degree of the remainder [tex]\( 3x \)[/tex] is less than the degree of the divisor [tex]\( x^3 - 3 \)[/tex].
- The quotient of the division is [tex]\( x + 5 \)[/tex].
Therefore, the quotient of [tex]\( \left(x^4 + 5x^3 - 3x - 15\right) \div \left(x^3 - 3\right) \)[/tex] is [tex]\( x + 5 \)[/tex], with a remainder of 0, meaning the division is exact.
