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Answer :
We want to divide
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
This can be done using polynomial long division.
1. Divide the leading terms:
The leading term of the dividend is [tex]$x^4$[/tex], and the leading term of the divisor is [tex]$x^3$[/tex]. Dividing these gives
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Thus, [tex]$x$[/tex] is the first term of the quotient.
2. Multiply and subtract:
Multiply the divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Now, subtract this product from the original dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x\\[1mm]
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
3. Repeat the process with the new polynomial:
The new polynomial is [tex]$5x^3 - 15$[/tex]. Divide its leading term [tex]$5x^3$[/tex] by the leading term of the divisor [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Therefore, add [tex]$5$[/tex] to the quotient.
4. Multiply and subtract again:
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]
The remainder is [tex]$0$[/tex], which means the division is exact.
5. Write the final result:
The quotient is the sum of the terms we found:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when dividing [tex]$x^4 + 5x^3 - 3x - 15$[/tex] by [tex]$x^3 - 3$[/tex] is
[tex]$$
\boxed{x+5}.
$$[/tex]
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
This can be done using polynomial long division.
1. Divide the leading terms:
The leading term of the dividend is [tex]$x^4$[/tex], and the leading term of the divisor is [tex]$x^3$[/tex]. Dividing these gives
[tex]$$
\frac{x^4}{x^3} = x.
$$[/tex]
Thus, [tex]$x$[/tex] is the first term of the quotient.
2. Multiply and subtract:
Multiply the divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Now, subtract this product from the original dividend:
[tex]$$
\begin{aligned}
\left(x^4 + 5x^3 - 3x - 15\right) - \left(x^4 - 3x\right)
&= x^4 + 5x^3 - 3x - 15 - x^4 + 3x\\[1mm]
&= 5x^3 - 15.
\end{aligned}
$$[/tex]
3. Repeat the process with the new polynomial:
The new polynomial is [tex]$5x^3 - 15$[/tex]. Divide its leading term [tex]$5x^3$[/tex] by the leading term of the divisor [tex]$x^3$[/tex]:
[tex]$$
\frac{5x^3}{x^3} = 5.
$$[/tex]
Therefore, add [tex]$5$[/tex] to the quotient.
4. Multiply and subtract again:
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]
The remainder is [tex]$0$[/tex], which means the division is exact.
5. Write the final result:
The quotient is the sum of the terms we found:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient when dividing [tex]$x^4 + 5x^3 - 3x - 15$[/tex] by [tex]$x^3 - 3$[/tex] is
[tex]$$
\boxed{x+5}.
$$[/tex]
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