The expression $$ p x^{3}-8 x^{2}+q x+6 $$ is exactly divisible by $$ x^{2}-2 x-3 $$ Find the values of $$ p $$ and $$ q $$.

Question

UpStudy AI · Accepted Answer

We are given that the polynomial

$$
p x^3 - 8 x^2 + q x + 6
$$

is exactly divisible by

$$
x^2 - 2x - 3.
$$

Since the divisor is of degree 2, its roots must also be roots of the cubic polynomial. First, factor the quadratic:

$$
x^2 - 2x - 3 = (x - 3)(x + 1).
$$

Thus, the roots are $$ x = 3 $$ and $$ x = -1 $$.

### Step 1: Substitute $$ x = 3 $$

Substitute into the polynomial:

$$
p(3)^3 - 8(3)^2 + q(3) + 6 = 0.
$$

Calculating each term:

$$
27p - 72 + 3q + 6 = 0.
$$

Simplify:

$$
27p + 3q - 66 = 0.
$$

Divide the entire equation by 3:

$$
9p + q - 22 = 0 \quad \Longrightarrow \quad 9p + q = 22.
$$

### Step 2: Substitute $$ x = -1 $$

Substitute $$ x = -1 $$ into the polynomial:

$$
p(-1)^3 - 8(-1)^2 + q(-1) + 6 = 0.
$$

Calculating each term:

$$
- p - 8 - q + 6 = 0.
$$

Simplify:

$$
- p - q - 2 = 0 \quad \Longrightarrow \quad p + q = -2.
$$

### Step 3: Solve the System of Equations

We now have the system:

$$
\begin{aligned}
9p + q &= 22, \
p + q &= -2.
\end{aligned}
$$

Subtract the second equation from the first:

$$
(9p + q) - (p + q) = 22 - (-2),
$$

$$
8p = 24 \quad \Longrightarrow \quad p = 3.
$$

Substitute $$ p = 3 $$ into $$ p + q = -2 $$:

$$
3 + q = -2 \quad \Longrightarrow \quad q = -5.
$$

### Final Answer

$$
\boxed{p = 3, \quad q = -5.}
$$
$$ p = 3 $$ and $$ q = -5 $$.

The expression \( p x^{3}-8 x^{2}+q x+6 \) is exactly divisible by \( x^{2}-2 x-3 \) Find the values of \( p \) and \( q \).