Math Problem Statement

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

Solution

Given the polynomial p(x)=2x3+ax2+bx+4p(x) = 2x^3 + ax^2 + bx + 4 with roots x=12,n,nx = \frac{1}{2}, n, -n, we can use Vieta's formulas to find the values of aa and bb.

Step 1: Expand the Polynomial in Terms of Roots

Since the roots of the polynomial are x=12,n,nx = \frac{1}{2}, n, -n, we can express p(x)p(x) as: p(x)=2(x12)(xn)(x+n)p(x) = 2 \left( x - \frac{1}{2} \right) (x - n)(x + n)

Simplifying (xn)(x+n)(x - n)(x + n):

(xn)(x+n)=x2n2(x - n)(x + n) = x^2 - n^2 So, p(x)=2(x12)(x2n2)p(x) = 2 \left( x - \frac{1}{2} \right) (x^2 - n^2)

Expanding 2(x12)(x2n2)2 \left( x - \frac{1}{2} \right) (x^2 - n^2):

p(x)=2(xx2xn212x2+n22)p(x) = 2 \left( x \cdot x^2 - x \cdot n^2 - \frac{1}{2} x^2 + \frac{n^2}{2} \right)

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Vieta's Formulas

Formulas

Vieta's formulas for polynomial roots
Expansion of polynomial (x - r1)(x - r2)(x - r3)

Theorems

Vieta's Theorem

Suitable Grade Level

Grades 10-12