Problem 44 - College Algebra

Question

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\)

Step-by-Step Solution

Answer

The roots of the given equation are -2, and the roots of the quadratic equation obtained from the division.

Step 1: Confirm the given root

We have been given that -2 is a root of the polynomial. We can check this by substituting -2 for x in the equation: \(f(-2) = 2*(-2)^3 - 3*(-2)^2 - 11*(-2) + 6\). If the result of this computation is zero, then -2 is indeed a root.

Step 2: Divide the Polynomial

We can find the other roots by performing polynomial division. Here, we will divide the polynomial \(2x^3 - 3x^2 - 11x + 6\) by the binomial \(x - (-2)\), which is \(x+2\). The quotient we get is another, simpler polynomial whose roots are also the roots of our original polynomial.

Step 3: Solving the Quotient Polynomial

Now, we solve the polynomial resulting from the division in step 2. This polynomial will be a quadratic equation which can be solved by factoring, completing the square, or using the quadratic formula.