Answer :
We want to divide
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
Here is the step-by-step division:
1. Set Up the Division:
We are looking for a polynomial quotient [tex]$Q(x)$[/tex] and a remainder [tex]$R(x)$[/tex] such that
[tex]$$
x^4 + 5x^3 - 3x - 15 = Q(x) \cdot (x^3 - 3) + R(x).
$$[/tex]
Since the degree of the numerator is [tex]$4$[/tex] and the degree of the denominator is [tex]$3$[/tex], the quotient will have degree [tex]$4-3=1$[/tex], so we expect a quotient of the form [tex]$ax+b$[/tex].
2. Divide the Leading Terms:
The leading term of the numerator is [tex]$x^4$[/tex], and the leading term of the denominator is [tex]$x^3$[/tex]. Dividing these gives
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
So, the first term of the quotient is [tex]$x$[/tex].
3. Multiply and Subtract:
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the original numerator:
[tex]$$
\begin{aligned}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
4. Repeat the Process:
Now, consider the new polynomial [tex]$5x^3 - 15$[/tex]. Its leading term is [tex]$5x^3$[/tex]. Divide this by the leading term of the divisor, [tex]$x^3$[/tex]:
[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 [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract from [tex]$5x^3 - 15$[/tex]:
[tex]$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$[/tex]
The remainder is [tex]$0$[/tex], meaning the division is exact.
6. Write the Final Answer:
Combining the terms obtained for the quotient gives
[tex]$$
Q(x) = x + 5.
$$[/tex]
Thus, the quotient of [tex]$\left(x^4 + 5x^3 - 3x - 15\right)$[/tex] by [tex]$\left(x^3 - 3\right)$[/tex] is
[tex]$$
\boxed{x + 5}.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
Here is the step-by-step division:
1. Set Up the Division:
We are looking for a polynomial quotient [tex]$Q(x)$[/tex] and a remainder [tex]$R(x)$[/tex] such that
[tex]$$
x^4 + 5x^3 - 3x - 15 = Q(x) \cdot (x^3 - 3) + R(x).
$$[/tex]
Since the degree of the numerator is [tex]$4$[/tex] and the degree of the denominator is [tex]$3$[/tex], the quotient will have degree [tex]$4-3=1$[/tex], so we expect a quotient of the form [tex]$ax+b$[/tex].
2. Divide the Leading Terms:
The leading term of the numerator is [tex]$x^4$[/tex], and the leading term of the denominator is [tex]$x^3$[/tex]. Dividing these gives
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
So, the first term of the quotient is [tex]$x$[/tex].
3. Multiply and Subtract:
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x \cdot (x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the original numerator:
[tex]$$
\begin{aligned}
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) &= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
4. Repeat the Process:
Now, consider the new polynomial [tex]$5x^3 - 15$[/tex]. Its leading term is [tex]$5x^3$[/tex]. Divide this by the leading term of the divisor, [tex]$x^3$[/tex]:
[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 [tex]$5$[/tex]:
[tex]$$
5 \cdot (x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract from [tex]$5x^3 - 15$[/tex]:
[tex]$$
(5x^3 - 15) - (5x^3 - 15) = 0.
$$[/tex]
The remainder is [tex]$0$[/tex], meaning the division is exact.
6. Write the Final Answer:
Combining the terms obtained for the quotient gives
[tex]$$
Q(x) = x + 5.
$$[/tex]
Thus, the quotient of [tex]$\left(x^4 + 5x^3 - 3x - 15\right)$[/tex] by [tex]$\left(x^3 - 3\right)$[/tex] is
[tex]$$
\boxed{x + 5}.
$$[/tex]
