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------------------------------------------------ The quotient of [tex]$\left(x^4+5x^3-3x-15\right)$[/tex] and [tex]$\left(x^3-3\right)$[/tex] is a polynomial. What is the quotient?

A. [tex]$x^7+5x^6-6x^4-30x^3+9x+45$[/tex]

B. [tex][tex]$x-5$[/tex][/tex]

C. [tex]$x+5$[/tex]

D. [tex]$x^7+5x^6+6x^4+30x^3+9x+45$[/tex]

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]).