Problem 73 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$2 x^{2}-2 x-112$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(2x^{2} - 2x - 112\) is \(-2(x - 7)(x - 8)\).

Step 1: Identify the Greatest Common Factor (GCF)

Observe the coefficients of the polynomial \(2x^{2} - 2x - 112\). Both 2 and 112 have a common factor, which is 2. However, the coefficient of the middle term, -2, is negative. This means that the GCF is -2.

Step 2: Factor out the GCF

Now, divide each term of the polynomial by -2. This results in \(-2(x^{2} + x + 56)\). It is clear that the polynomial within the brackets is a quadratic polynomial and it can be factored.

Step 3: Factorize the Quadratic Polynomial

To factorize \(x^{2} + x + 56\), find two numbers that multiply to 56 and add to 1 (from the x's coefficient). These numbers are -7 and -8. Substituting these values, the expression becomes \(x^{2} - 7x - 8x + 56\). This can be further factored into \(x(x - 7) - 8(x - 7)\).

Step 4: Combine Like Terms

Combine the like terms we have from step 3. We combine \((x - 7)(x - 8)\), and adjust the GCF that we factored out in step 2, \( -2(x - 7)(x - 8)\) is our completely factored polynomial.