Answer :
To find the quotient of the given polynomials [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] divided by [tex]\( (x^3 - 3) \)[/tex], we perform polynomial division. Here are the steps:
1. Setup the division: Write down the dividend [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] and the divisor [tex]\( (x^3 - 3) \)[/tex].
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]. Dividing these gives [tex]\( x \)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\( (x^3 - 3) \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Repeat the process:
- The new leading term of the remainder is [tex]\( 5x^3 \)[/tex]. Divide it by the leading term of the divisor [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the divisor [tex]\( (x^3 - 3) \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract this from the current remainder:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- The division yields a zero remainder, showing the division is exact.
- Therefore, the quotient is [tex]\( x + 5 \)[/tex].
The quotient of the polynomial division is [tex]\( x + 5 \)[/tex].
1. Setup the division: Write down the dividend [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] and the divisor [tex]\( (x^3 - 3) \)[/tex].
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]. Dividing these gives [tex]\( x \)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\( (x^3 - 3) \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Repeat the process:
- The new leading term of the remainder is [tex]\( 5x^3 \)[/tex]. Divide it by the leading term of the divisor [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the divisor [tex]\( (x^3 - 3) \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract this from the current remainder:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
5. Conclusion:
- The division yields a zero remainder, showing the division is exact.
- Therefore, the quotient is [tex]\( x + 5 \)[/tex].
The quotient of the polynomial division is [tex]\( x + 5 \)[/tex].
