Answer :
To find the quotient of
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
divided by
[tex]$$
x^3 - 3,
$$[/tex]
we can perform polynomial long division. Here is the step-by-step process:
1. First Division Step:
Divide the leading term of the dividend, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex], to get the first term of the quotient:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Now, multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the original dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
2. Second Division Step:
Now, take the new dividend [tex]$5x^3 - 15$[/tex] and divide its leading term by the leading term of the divisor:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by [tex]$5$[/tex]:
[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]
Since the remainder is [tex]$0$[/tex], the division process is complete.
3. Conclusion:
The quotient obtained from the division is the sum of the terms found in each step:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when dividing
[tex]$$
x^4 + 5x^3 - 3x - 15 \quad \text{by} \quad x^3 - 3
$$[/tex]
is
[tex]$$
\boxed{x+5}.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
divided by
[tex]$$
x^3 - 3,
$$[/tex]
we can perform polynomial long division. Here is the step-by-step process:
1. First Division Step:
Divide the leading term of the dividend, [tex]$x^4$[/tex], by the leading term of the divisor, [tex]$x^3$[/tex], to get the first term of the quotient:
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Now, multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the original dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x \\
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
2. Second Division Step:
Now, take the new dividend [tex]$5x^3 - 15$[/tex] and divide its leading term by the leading term of the divisor:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Multiply the divisor by [tex]$5$[/tex]:
[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]
Since the remainder is [tex]$0$[/tex], the division process is complete.
3. Conclusion:
The quotient obtained from the division is the sum of the terms found in each step:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when dividing
[tex]$$
x^4 + 5x^3 - 3x - 15 \quad \text{by} \quad x^3 - 3
$$[/tex]
is
[tex]$$
\boxed{x+5}.
$$[/tex]
