Question

Factor each polynomial. $$ x^{2}+2 x-8 $$

Short answer

(x - 2)(x + 4)

Step-by-step solution

Identify the polynomial form

The given polynomial is in the form of a quadratic equation: \[ ax^2 + bx + c \] where the coefficients are: \[ a = 1, b = 2, c = -8 \]

Find the factors of the constant term

List all pairs of factors of the constant term \( c = -8 \): \[ (-1, 8), (1, -8), (-2, 4), (2, -4) \]

Determine the pair that sums to the middle coefficient

We need to find a pair from Step 2 whose sum is equal to the middle coefficient \( b = 2 \)Checking the pairs: \[ -1 + 8 = 7 \]\[ 1 - 8 = -7 \]\[ -2 + 4 = 2 \] So the correct pair is \( -2 \text{ and } 4 \)

Rewrite the polynomial with the found factors

Rewrite the polynomial \( x^2 + 2x - 8 \) in the factored form based on the pair found in Step 3:\[ x^2 + 2x - 8 = (x - 2)(x + 4) \]

Verify the factorization

To ensure the factorization is correct, expand \( (x - 2)(x + 4) \): \[ (x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8 \] The original polynomial is obtained, which verifies the factorization.

Key concepts

Quadratic Equations

Quadratic equations are a type of polynomial where the highest power of the variable is 2. The general form of a quadratic equation is \( ax^2 + bx + c \). In this form, \( a \),\( b \), and \( c \) are coefficients and \( x \) is the variable. Quadratic equations often appear in different areas of mathematics and can be solved using various methods like factoring, completing the square, or using the quadratic formula. In the exercise, we determined that the given polynomial \( x^2 + 2x - 8 \) is a quadratic equation and identified the coefficients. Knowing the structure helps in moving forward with the factorization.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. For quadratic equations, our goal is to express \( ax^2 + bx + c \) as \( (px + q)(rx + s) \). Factoring helps in simplifying equations and solving them. Here’s how we did it in the exercise:

• We listed pairs of factors of the constant term \( c = -8 \) which are (-1, 8), (1, -8), (-2, 4), and (2, -4).


• Next, we determined which pair sums to the middle coefficient \( b = 2 \), in this case, \( -2 + 4 = 2 \).


• Using the pair, we rewrote \( x^2 + 2x - 8 \) as \( (x - 2)(x + 4) \).


• To confirm our factorization, we expanded \( (x - 2)(x + 4) \) and got back the original polynomial.

This step-by-step approach ensures accuracy in factorization and deepens understanding.

Coefficients

Coefficients are numbers that multiply the variables in a polynomial. In a quadratic equation \( ax^2 + bx + c \), the coefficients are:

• \( a \): This is the coefficient of \( x^2 \) and in our example, \( a = 1 \).


• \( b \): This is the coefficient of \( x \) and in our example, \( b = 2 \).


• \( c \): This is the constant term or the coefficient of \( x^0 \), which is simply \( c = -8 \).

Recognizing and correctly identifying these coefficients is crucial when solving quadratic equations.
In our exercise, recognizing these values allowed us to list possible factor pairs and identify the pair that summed to the middle coefficient \( b \), leading us to the correct factorization.