Given that (x-1) and (2x+ 1) is a factor of px^3+ qx^2+ qx - 2
, find p and g.

We are given that \((x - 1)\) and \((2x + 1)\) are factors of the polynomial \(p x^3 + q x^2 + q x - 2\). We need to determine the values of \(p\) and \(q\).

### Step 1: Factor Form of the Polynomial

Since \((x - 1)\) and \((2x + 1)\) are factors of the cubic polynomial, the general factorized form of the polynomial can be written as:

\[
p(x - 1)(2x + 1)(x + r)
\]
where \(r\) is a constant, and \(p\) is the leading coefficient.

### Step 2: Expand the Factors

First, expand \((x - 1)(2x + 1)\):

\[
(x - 1)(2x + 1) = x(2x + 1) - 1(2x + 1) = 2x^2 + x - 2x - 1 = 2x^2 - x - 1
\]

Now, multiply this result by \((x + r)\):

\[
(2x^2 - x - 1)(x + r) = (2x^2 - x - 1)x + (2x^2 - x - 1)r
\]
\[
= 2x^3 - x^2 - x - rx + 2r x^2 - r x - r
\]
\[
= 2x^3 + (2r - 1)x^2 + (-r - 1)x - r
\]

### Step 3: Compare with the Given Polynomial

We now compare this expanded form with the given polynomial \(p x^3 + q x^2 + q x - 2\):

\[
p(2x^3 + (2r - 1)x^2 + (-r - 1)x - r) = px^3 + qx^2 + qx - 2
\]

By comparing the coefficients of the corresponding powers of \(x\), we get the following system of equations:

1. \(p \cdot 2 = p \Rightarrow 2p = p \Rightarrow p = 1\)
2. \(p(2r - 1) = q \Rightarrow (2r - 1) = q\)
3. \(p(-r - 1) = q \Rightarrow (-r - 1) = q\)
4. \(-p \cdot r = -2 \Rightarrow -r = -2 \Rightarrow r = 2\)

### Step 4: Solve for \(q\)

Substitute \(r = 2\) into the equations for \(q\):

\[
2r - 1 = q \Rightarrow 2(2) - 1 = q \Rightarrow 4 - 1 = q \Rightarrow q = 3
\]
\[
-r - 1 = q \Rightarrow -(2) - 1 = q \Rightarrow -2 - 1 = q \Rightarrow q = -3
\]

We get \(p = 1\) and \(q = 3\).

### Final Answer:
\[
p = 1, \quad q = 3
\]

Let me know if you'd like further details!

Here are 5 related questions:
1. How would the solution change if the polynomial had degree 4?
2. Can you explain how the Factor Theorem works in this problem?
3. What happens if one of the factors is not linear, like \((x^2 + 1)\)?
4. How do we handle a scenario where the polynomial has complex roots?
5. What if the leading coefficient \(p\) was unknown?

**Tip**: When expanding factored polynomials, grouping terms carefully helps avoid mistakes in comparing coefficients!

Given that (x-1) and (2x+ 1) is a factor of px^3+ qx^2+ qx - 2, find p and q.

Explore how to factorize the cubic polynomial px^3 + qx^2 + qx - 2, given that (x - 1) and (2x + 1) are factors. This solution uses polynomial factorization and coefficient comparison to find p and q. Suitable for students in Grades 10-12.

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