Answer :
We want to divide
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
Because the degree of the dividend (which is 4) is exactly one more than the degree of the divisor (which is 3), we expect the quotient to be a linear polynomial. Let's assume the quotient is of the form
[tex]$$
q(x)=Ax+B.
$$[/tex]
Multiplying the divisor by this assumed quotient gives:
[tex]$$
(Ax + B)(x^3 - 3) = Ax^4 + Bx^3 - 3Ax - 3B.
$$[/tex]
For this product to be identical to the dividend, the coefficients of corresponding powers of [tex]$x$[/tex] must be equal. Hence, we compare:
[tex]\[
\begin{array}{rcl}
\text{Coefficient of } x^4: & A &= 1, \\
\text{Coefficient of } x^3: & B &= 5, \\
\text{Coefficient of } x: & -3A &= -3, \quad \text{which is satisfied since } A=1, \\
\text{Constant term:} & -3B &= -15, \quad \text{which is satisfied since } B=5. \\
\end{array}
\][/tex]
Since all these equations are consistent, the quotient is:
[tex]$$
q(x) = x + 5.
$$[/tex]
Also, since the multiplication reproduces exactly the original dividend, the remainder is [tex]$0$[/tex].
Thus, the quotient when dividing
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3
$$[/tex]
is:
[tex]$$
x+5.
$$[/tex]
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3.
$$[/tex]
Because the degree of the dividend (which is 4) is exactly one more than the degree of the divisor (which is 3), we expect the quotient to be a linear polynomial. Let's assume the quotient is of the form
[tex]$$
q(x)=Ax+B.
$$[/tex]
Multiplying the divisor by this assumed quotient gives:
[tex]$$
(Ax + B)(x^3 - 3) = Ax^4 + Bx^3 - 3Ax - 3B.
$$[/tex]
For this product to be identical to the dividend, the coefficients of corresponding powers of [tex]$x$[/tex] must be equal. Hence, we compare:
[tex]\[
\begin{array}{rcl}
\text{Coefficient of } x^4: & A &= 1, \\
\text{Coefficient of } x^3: & B &= 5, \\
\text{Coefficient of } x: & -3A &= -3, \quad \text{which is satisfied since } A=1, \\
\text{Constant term:} & -3B &= -15, \quad \text{which is satisfied since } B=5. \\
\end{array}
\][/tex]
Since all these equations are consistent, the quotient is:
[tex]$$
q(x) = x + 5.
$$[/tex]
Also, since the multiplication reproduces exactly the original dividend, the remainder is [tex]$0$[/tex].
Thus, the quotient when dividing
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3
$$[/tex]
is:
[tex]$$
x+5.
$$[/tex]
