Answer :
We want to divide
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
Since the degree of the dividend is 4 and the degree of the divisor is 3, the quotient will be a polynomial of degree [tex]$4-3=1$[/tex], meaning it has the form
[tex]$$
ax + b.
$$[/tex]
To find [tex]$a$[/tex] and [tex]$b$[/tex], we write the product of the divisor and the assumed quotient:
[tex]$$
(x^3 - 3)(ax + b) = ax^4 + bx^3 - 3ax - 3b.
$$[/tex]
This product must equal the dividend:
[tex]$$
ax^4 + bx^3 - 3ax - 3b = x^4 + 5x^3 - 3x - 15.
$$[/tex]
Now, we equate the coefficients for the corresponding powers of [tex]$x$[/tex]:
1. Coefficient of [tex]$x^4$[/tex]:
[tex]$$
a = 1.
$$[/tex]
2. Coefficient of [tex]$x^3$[/tex]:
[tex]$$
b = 5.
$$[/tex]
3. Coefficient of [tex]$x$[/tex]:
[tex]$$
-3a = -3 \quad \text{(which is true since } a=1\text{)}.
$$[/tex]
4. Constant term:
[tex]$$
-3b = -15 \quad \text{(which is true since } b=5\text{)}.
$$[/tex]
Since all the coefficients match, the quotient is
[tex]$$
\boxed{x + 5},
$$[/tex]
and the division is exact (the remainder is [tex]$0$[/tex]).
[tex]$$
x^4 + 5x^3 - 3x - 15
$$[/tex]
by
[tex]$$
x^3 - 3.
$$[/tex]
Since the degree of the dividend is 4 and the degree of the divisor is 3, the quotient will be a polynomial of degree [tex]$4-3=1$[/tex], meaning it has the form
[tex]$$
ax + b.
$$[/tex]
To find [tex]$a$[/tex] and [tex]$b$[/tex], we write the product of the divisor and the assumed quotient:
[tex]$$
(x^3 - 3)(ax + b) = ax^4 + bx^3 - 3ax - 3b.
$$[/tex]
This product must equal the dividend:
[tex]$$
ax^4 + bx^3 - 3ax - 3b = x^4 + 5x^3 - 3x - 15.
$$[/tex]
Now, we equate the coefficients for the corresponding powers of [tex]$x$[/tex]:
1. Coefficient of [tex]$x^4$[/tex]:
[tex]$$
a = 1.
$$[/tex]
2. Coefficient of [tex]$x^3$[/tex]:
[tex]$$
b = 5.
$$[/tex]
3. Coefficient of [tex]$x$[/tex]:
[tex]$$
-3a = -3 \quad \text{(which is true since } a=1\text{)}.
$$[/tex]
4. Constant term:
[tex]$$
-3b = -15 \quad \text{(which is true since } b=5\text{)}.
$$[/tex]
Since all the coefficients match, the quotient is
[tex]$$
\boxed{x + 5},
$$[/tex]
and the division is exact (the remainder is [tex]$0$[/tex]).
