Answer :
To find the quotient of the polynomial division between [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] and [tex]\( x^3 - 3 \)[/tex], we can perform polynomial long division. Here's how it works step-by-step:
1. Setup the Division:
- We are dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [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 the entire divisor [tex]\( x^3 - 3 \)[/tex] by the result from the previous division, [tex]\( x \)[/tex], giving [tex]\( x(x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex], resulting in:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Repeat the Process:
- Now, divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get the next term of the quotient: [tex]\( 5x^3 \div x^3 = 5 \)[/tex].
- Multiply the whole divisor [tex]\( x^3 - 3 \)[/tex] by this new result [tex]\( 5 \)[/tex], giving [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from the current dividend [tex]\( 5x^3 - 15 \)[/tex] to get:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- Since there are no more terms remaining, the division is complete.
- The quotient of this division is [tex]\( x + 5 \)[/tex].
- There is no remainder.
Thus, the quotient of [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] divided by [tex]\( (x^3 - 3) \)[/tex] is [tex]\( x + 5 \)[/tex].
1. Setup the Division:
- We are dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [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 the entire divisor [tex]\( x^3 - 3 \)[/tex] by the result from the previous division, [tex]\( x \)[/tex], giving [tex]\( x(x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex], resulting in:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Repeat the Process:
- Now, divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get the next term of the quotient: [tex]\( 5x^3 \div x^3 = 5 \)[/tex].
- Multiply the whole divisor [tex]\( x^3 - 3 \)[/tex] by this new result [tex]\( 5 \)[/tex], giving [tex]\( 5(x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from the current dividend [tex]\( 5x^3 - 15 \)[/tex] to get:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- Since there are no more terms remaining, the division is complete.
- The quotient of this division is [tex]\( x + 5 \)[/tex].
- There is no remainder.
Thus, the quotient of [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] divided by [tex]\( (x^3 - 3) \)[/tex] is [tex]\( x + 5 \)[/tex].
