Answer :
We start by dividing the polynomial
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
Step 1: Divide the leading terms
The highest degree term in the dividend is [tex]$x^4$[/tex], and in the divisor it 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].
Step 2: Multiply and subtract
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the dividend:
[tex]\[
\begin{array}{rcl}
x^4 + 5x^3 - 3x - 15 & - & (x^4 - 3x) \\
\hline
& = & 5x^3 - 15.
\end{array}
\][/tex]
Step 3: Repeat the process
Now, we take the new dividend [tex]$5x^3 - 15$[/tex]. Divide the 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]
This gives the next term of the quotient, which is [tex]$5$[/tex].
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5(x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0.
\][/tex]
The remainder is [tex]$0$[/tex], which means the division is exact.
Step 4: Write the quotient
The quotient obtained is the sum of the terms calculated:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient of the division of [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]
Step 1: Divide the leading terms
The highest degree term in the dividend is [tex]$x^4$[/tex], and in the divisor it 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].
Step 2: Multiply and subtract
Multiply the entire divisor by [tex]$x$[/tex]:
[tex]$$
x(x^3 - 3) = x^4 - 3x.
$$[/tex]
Subtract this product from the dividend:
[tex]\[
\begin{array}{rcl}
x^4 + 5x^3 - 3x - 15 & - & (x^4 - 3x) \\
\hline
& = & 5x^3 - 15.
\end{array}
\][/tex]
Step 3: Repeat the process
Now, we take the new dividend [tex]$5x^3 - 15$[/tex]. Divide the 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]
This gives the next term of the quotient, which is [tex]$5$[/tex].
Multiply the divisor by [tex]$5$[/tex]:
[tex]$$
5(x^3 - 3) = 5x^3 - 15.
$$[/tex]
Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0.
\][/tex]
The remainder is [tex]$0$[/tex], which means the division is exact.
Step 4: Write the quotient
The quotient obtained is the sum of the terms calculated:
[tex]$$
x + 5.
$$[/tex]
Thus, the quotient of the division of [tex]$x^4+5x^3-3x-15$[/tex] by [tex]$x^3-3$[/tex] is
[tex]$$
\boxed{x + 5}.
$$[/tex]
