p(x) = 2x^3 + ax^2 + bx + 4    roots of this equation are x=1/2, n, -n, find a and b

Given the polynomial \( p(x) = 2x^3 + ax^2 + bx + 4 \) with roots \( x = \frac{1}{2}, n, -n \), we can use Vieta's formulas to find the values of \( a \) and \( b \).

### Step 1: Expand the Polynomial in Terms of Roots

Since the roots of the polynomial are \( x = \frac{1}{2}, n, -n \), we can express \( p(x) \) as:
\[
p(x) = 2 \left( x - \frac{1}{2} \right) (x - n)(x + n)
\]

#### Simplifying \( (x - n)(x + n) \):
\[
(x - n)(x + n) = x^2 - n^2
\]
So,
\[
p(x) = 2 \left( x - \frac{1}{2} \right) (x^2 - n^2)
\]

#### Expanding \( 2 \left( x - \frac{1}{2} \right) (x^2 - n^2) \):
\[
p(x) = 2 \left( x \cdot x^2 - x \cdot n^2 - \frac{1}{2} x^2 + \frac{n^2}{2} \right)
\]


p(x) = 2x^3 + ax^2 + bx + 4; roots of this equation are x = 1/2, n, -n. Find a and b.

Explore how to determine coefficients a and b in the polynomial p(x) = 2x^3 + ax^2 + bx + 4, given roots x = 1/2, n, -n. This problem involves polynomial expansion, Vieta's formulas, and algebraic manipulation, making it suitable for Grades 10-12.

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