Answer :
To find the quotient of the polynomial [tex]\( f(x) = x^4 + 5x^3 - 3x - 15 \)[/tex] by the divisor [tex]\( d(x) = x^3 - 3 \)[/tex], we perform polynomial division.
### Step-by-Step Solution:
1. Setup:
We have two polynomials: the dividend [tex]\( f(x) = x^4 + 5x^3 - 3x - 15 \)[/tex] and the divisor [tex]\( d(x) = x^3 - 3 \)[/tex].
2. Divide the Leading Terms:
The leading term of [tex]\( f(x) \)[/tex] is [tex]\( x^4 \)[/tex] and the leading term of [tex]\( d(x) \)[/tex] is [tex]\( x^3 \)[/tex]. We divide the leading terms:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract:
We multiply the entire divisor [tex]\( d(x) \)[/tex] by this first term of the quotient and subtract the result from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^3 - 3) \cdot x = x^4 - 3x
\][/tex]
Subtract this product from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat the Process:
Now, the new dividend is [tex]\( 5x^3 - 15 \)[/tex]. We focus again on the leading terms:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term in the quotient is [tex]\( 5 \)[/tex].
5. Multiply and Subtract Again:
Multiply the divisor by this new part of the quotient:
[tex]\[
(x^3 - 3) \cdot 5 = 5x^3 - 15
\][/tex]
Subtract this product from the new dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and the quotient is fully derived by combining the terms we found at each step, the quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is:
[tex]\[
x + 5
\][/tex]
Thus, the quotient is:
[tex]\[
\boxed{x + 5}
\][/tex]
### Step-by-Step Solution:
1. Setup:
We have two polynomials: the dividend [tex]\( f(x) = x^4 + 5x^3 - 3x - 15 \)[/tex] and the divisor [tex]\( d(x) = x^3 - 3 \)[/tex].
2. Divide the Leading Terms:
The leading term of [tex]\( f(x) \)[/tex] is [tex]\( x^4 \)[/tex] and the leading term of [tex]\( d(x) \)[/tex] is [tex]\( x^3 \)[/tex]. We divide the leading terms:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract:
We multiply the entire divisor [tex]\( d(x) \)[/tex] by this first term of the quotient and subtract the result from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^3 - 3) \cdot x = x^4 - 3x
\][/tex]
Subtract this product from [tex]\( f(x) \)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat the Process:
Now, the new dividend is [tex]\( 5x^3 - 15 \)[/tex]. We focus again on the leading terms:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term in the quotient is [tex]\( 5 \)[/tex].
5. Multiply and Subtract Again:
Multiply the divisor by this new part of the quotient:
[tex]\[
(x^3 - 3) \cdot 5 = 5x^3 - 15
\][/tex]
Subtract this product from the new dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and the quotient is fully derived by combining the terms we found at each step, the quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex] is:
[tex]\[
x + 5
\][/tex]
Thus, the quotient is:
[tex]\[
\boxed{x + 5}
\][/tex]
