Answer :
To solve the problem of finding the quotient of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex], we use polynomial long division. Here's a detailed step-by-step breakdown of the process:
1. Set Up the Division:
- The dividend is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The divisor 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].
- This gives you the first term of the quotient: [tex]\( x^4 / x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which gives [tex]\( x^4 - 3x \)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x -15
\][/tex]
4. Repeat the Process:
- Repeat the division process by dividing the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]. This gives [tex]\( 5 \)[/tex].
- Multiply and subtract:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Combine Results:
- Since there is nothing left to bring down and the subtraction results in zero, the division is complete.
- The quotient of the division is the expression we obtained: [tex]\( x + 5 \)[/tex].
Therefore, the quotient of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex].
1. Set Up the Division:
- The dividend is [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
- The divisor 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].
- This gives you the first term of the quotient: [tex]\( x^4 / x^3 = x \)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which gives [tex]\( x^4 - 3x \)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x -15
\][/tex]
4. Repeat the Process:
- Repeat the division process by dividing the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]. This gives [tex]\( 5 \)[/tex].
- Multiply and subtract:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. Combine Results:
- Since there is nothing left to bring down and the subtraction results in zero, the division is complete.
- The quotient of the division is the expression we obtained: [tex]\( x + 5 \)[/tex].
Therefore, the quotient of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex].
