Answer :
To solve this problem, we need to find the quotient obtained when dividing the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
Here is the step-by-step solution:
1. Set up the division:
We're dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
2. Look at the leading terms:
- The leading term of the dividend is [tex]\( x^4 \)[/tex].
- The leading term of the divisor is [tex]\( x^3 \)[/tex].
3. Divide the leading terms:
Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( x \)[/tex].
4. Multiply and subtract:
Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
(x)(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
5. Repeat the process:
- Now, our new dividend is [tex]\( 5x^3 - 15 \)[/tex].
6. Divide again:
This time, divide [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( 5 \)[/tex].
7. Multiply and subtract:
Multiply [tex]\( 5 \)[/tex] by the divisor:
[tex]\[
(5)(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from [tex]\( 5x^3 - 15 \)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
8. Conclusion:
The division results in a quotient of [tex]\( x + 5 \)[/tex] with a remainder of 0.
So, the quotient of dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex]. Therefore, the correct option is [tex]\( x + 5 \)[/tex].
Here is the step-by-step solution:
1. Set up the division:
We're dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
2. Look at the leading terms:
- The leading term of the dividend is [tex]\( x^4 \)[/tex].
- The leading term of the divisor is [tex]\( x^3 \)[/tex].
3. Divide the leading terms:
Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( x \)[/tex].
4. Multiply and subtract:
Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
(x)(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
5. Repeat the process:
- Now, our new dividend is [tex]\( 5x^3 - 15 \)[/tex].
6. Divide again:
This time, divide [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( 5 \)[/tex].
7. Multiply and subtract:
Multiply [tex]\( 5 \)[/tex] by the divisor:
[tex]\[
(5)(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from [tex]\( 5x^3 - 15 \)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
8. Conclusion:
The division results in a quotient of [tex]\( x + 5 \)[/tex] with a remainder of 0.
So, the quotient of dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex]. Therefore, the correct option is [tex]\( x + 5 \)[/tex].
