Answer :
We want to divide the polynomial
[tex]$$P(x)=x^4+5x^3-3x-15$$[/tex]
by
[tex]$$D(x)=x^3-3.$$[/tex]
Since the degree of [tex]$P(x)$[/tex] is [tex]$4$[/tex] and the degree of [tex]$D(x)$[/tex] is [tex]$3$[/tex], the quotient will be a linear polynomial (of degree [tex]$1$[/tex]).
Step 1. Divide the leading term of [tex]$P(x)$[/tex] by the leading term of [tex]$D(x)$[/tex]
The leading term of [tex]$P(x)$[/tex] is [tex]$x^4$[/tex], and the leading term of [tex]$D(x)$[/tex] is [tex]$x^3$[/tex].
Divide:
[tex]$$\frac{x^4}{x^3} = x.$$[/tex]
This gives the first term of the quotient.
Step 2. Multiply the divisor by the first term of the quotient
Multiply [tex]$D(x)$[/tex] by [tex]$x$[/tex]:
[tex]$$x(x^3-3)=x^4-3x.$$[/tex]
Step 3. Subtract the result from [tex]$P(x)$[/tex]
Subtract the product from [tex]$P(x)$[/tex]:
[tex]\[
\begin{array}{rcl}
x^4+5x^3-3x-15 & - & \left(x^4-3x\right) \\
& = & x^4+5x^3-3x-15 - x^4+3x \\
& = & 5x^3-15.
\end{array}
\][/tex]
Step 4. Divide the new leading term by the leading term of the divisor
Now, divide the new leading term [tex]$5x^3$[/tex] by [tex]$x^3$[/tex]:
[tex]$$\frac{5x^3}{x^3}= 5.$$[/tex]
This is the second term of the quotient.
Step 5. Multiply the divisor by this new term and subtract
Multiply [tex]$D(x)$[/tex] by [tex]$5$[/tex]:
[tex]$$5(x^3-3)=5x^3-15.$$[/tex]
Subtract:
[tex]\[
\begin{array}{rcl}
5x^3-15 & - & \left(5x^3-15\right) \\
& = & 0.
\end{array}
\][/tex]
There is no remainder.
Step 6. Write the final quotient
The quotient from the division is the sum of the terms we obtained:
[tex]$$Q(x)=x+5.$$[/tex]
Thus, the quotient of [tex]$\left(x^4+5 x^3-3 x-15\right)$[/tex] divided by [tex]$\left(x^3-3\right)$[/tex] is
[tex]$$\boxed{x+5}.$$[/tex]
[tex]$$P(x)=x^4+5x^3-3x-15$$[/tex]
by
[tex]$$D(x)=x^3-3.$$[/tex]
Since the degree of [tex]$P(x)$[/tex] is [tex]$4$[/tex] and the degree of [tex]$D(x)$[/tex] is [tex]$3$[/tex], the quotient will be a linear polynomial (of degree [tex]$1$[/tex]).
Step 1. Divide the leading term of [tex]$P(x)$[/tex] by the leading term of [tex]$D(x)$[/tex]
The leading term of [tex]$P(x)$[/tex] is [tex]$x^4$[/tex], and the leading term of [tex]$D(x)$[/tex] is [tex]$x^3$[/tex].
Divide:
[tex]$$\frac{x^4}{x^3} = x.$$[/tex]
This gives the first term of the quotient.
Step 2. Multiply the divisor by the first term of the quotient
Multiply [tex]$D(x)$[/tex] by [tex]$x$[/tex]:
[tex]$$x(x^3-3)=x^4-3x.$$[/tex]
Step 3. Subtract the result from [tex]$P(x)$[/tex]
Subtract the product from [tex]$P(x)$[/tex]:
[tex]\[
\begin{array}{rcl}
x^4+5x^3-3x-15 & - & \left(x^4-3x\right) \\
& = & x^4+5x^3-3x-15 - x^4+3x \\
& = & 5x^3-15.
\end{array}
\][/tex]
Step 4. Divide the new leading term by the leading term of the divisor
Now, divide the new leading term [tex]$5x^3$[/tex] by [tex]$x^3$[/tex]:
[tex]$$\frac{5x^3}{x^3}= 5.$$[/tex]
This is the second term of the quotient.
Step 5. Multiply the divisor by this new term and subtract
Multiply [tex]$D(x)$[/tex] by [tex]$5$[/tex]:
[tex]$$5(x^3-3)=5x^3-15.$$[/tex]
Subtract:
[tex]\[
\begin{array}{rcl}
5x^3-15 & - & \left(5x^3-15\right) \\
& = & 0.
\end{array}
\][/tex]
There is no remainder.
Step 6. Write the final quotient
The quotient from the division is the sum of the terms we obtained:
[tex]$$Q(x)=x+5.$$[/tex]
Thus, the quotient of [tex]$\left(x^4+5 x^3-3 x-15\right)$[/tex] divided by [tex]$\left(x^3-3\right)$[/tex] is
[tex]$$\boxed{x+5}.$$[/tex]
