Answer :
Let's solve the problem step by step:
We need to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by the polynomial [tex]\( x^3 - 3 \)[/tex].
1. Division Setup:
Begin the polynomial long division by writing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] as the dividend and [tex]\( x^3 - 3 \)[/tex] as the divisor.
2. First Term:
- Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex].
- This gives you [tex]\( x^4 \div x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Next Term:
- The new dividend is [tex]\( 5x^3 + 0x - 15 \)[/tex].
- Divide the leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], resulting in [tex]\( 5 \)[/tex].
5. Multiply and Subtract Again:
- Multiply [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract:
[tex]\[
(5x^3 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
The quotient from the division is [tex]\( x + 5 \)[/tex].
So, the quotient of dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex], which matches the given option "x + 5".
We need to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by the polynomial [tex]\( x^3 - 3 \)[/tex].
1. Division Setup:
Begin the polynomial long division by writing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] as the dividend and [tex]\( x^3 - 3 \)[/tex] as the divisor.
2. First Term:
- Divide the leading term of the dividend [tex]\( x^4 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex].
- This gives you [tex]\( x^4 \div x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
4. Next Term:
- The new dividend is [tex]\( 5x^3 + 0x - 15 \)[/tex].
- Divide the leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], resulting in [tex]\( 5 \)[/tex].
5. Multiply and Subtract Again:
- Multiply [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract:
[tex]\[
(5x^3 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
The quotient from the division is [tex]\( x + 5 \)[/tex].
So, the quotient of dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex], which matches the given option "x + 5".
