Problem 67 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$4 x^{2}-4 x-24$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(4x^2-4x-24\) is \(4(x-3)(x+2)\).

Step 1: Identify the Polynomial

The polynomial given in the problem is \(4x^2-4x-24\). We want to factor this polynomial completely.

Step 2: Factor out the Greatest Common Factor

First, we notice that there's a common factor of 4 in each term of the polynomial. Factoring out the 4 gives us: \(4(x^2-x-6)\).

Step 3: Factor the Quadratic Polynomial

Next, we need to factor the quadratic polynomial \(x^2-x-6\). The factors of -6 that add up to -1 are -3 and 2. Although we normally write the constant term (2) first, in this case, we are looking for the factors of a negative number, so we write -3 first. Consequently, this gives us \(x^2-x-6=(x-3)(x+2)\). 2 and -3 are the roots of the equation \(x^2-x-6=0\).

Step 4: Write the completely factored polynomial

Finally, we substitute the value factored in step 2 into step 3 and write the completely factored polynomial as \(4x^2-4x-24=4(x-3)(x+2)\).