Question

Factor each polynomial. $$ x^{2}+6 x+8 $$

Short answer

\((x + 2)(x + 4)\)

Step-by-step solution

Identify the Polynomial Structure

The given polynomial is a quadratic trinomial of the form: \[ax^2 + bx + c\] For this exercise, \[a = 1, b = 6, c = 8\]

Determine Factor Pairs of c

List the factor pairs of the constant term (c), 8. The pairs are: 1 and 8, 2 and 4.

Find the Pair that Adds to b

Select the pair of factors that add up to the middle coefficient (b), 6. Since 2 + 4 = 6, the required pair is 2 and 4.

Express the Polynomial as Binomials

Using the factor pair (2, 4), express the polynomial as a product of two binomials: \[(x + 2)(x + 4)\]

Verify the Factorization

Expand the binomials to ensure correctness: \[(x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8\]Since the expanded form matches the original polynomial, the factorization is correct.

Key concepts

quadratic trinomial

Understanding a quadratic trinomial is crucial in factoring quadratic polynomials. A quadratic trinomial is a polynomial with three terms and the highest degree of 2. The standard form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this exercise, the polynomial \(x^2 + 6x + 8\) fits this form with \(a = 1\), \(b = 6\), and \(c = 8\). Knowing these values helps us identify the next steps in factoring.

factor pairs

Once we have identified \(a\), \(b\), and \(c\) in the quadratic trinomial \(ax^2 + bx + c\), our next task is to find factor pairs of the constant term \(c\). Factor pairs are two numbers that multiply to give the value of \(c\). For \(c = 8\), the factor pairs are:

• 1 and 8
• 2 and 4

Listing these factor pairs is essential because we need to find the pair that adds up to the middle coefficient \(b\) to move forward with factoring the trinomial.

binomial

A binomial is a polynomial with exactly two terms. In the context of factoring a quadratic trinomial, once we identify the correct factor pair that adds up to \(b\), we express the quadratic polynomial as a product of two binomials. For the polynomial \(x^2 + 6x + 8\), the factor pair (2, 4) adds up to 6. Therefore, we write the original trinomial as a product of two binomials: \((x + 2)(x + 4)\).

polynomial expansion

Expanding a polynomial involves multiplying the binomials back together to verify our factorization is correct. This is done by distributing the terms in the binomials and combining like terms. For our example, we expand \((x + 2)(x + 4)\) as follows:

- Multiply \(x\) by each term in the second binomial: \(x \times x = x^2\) and \(x \times 4 = 4x\)
- Multiply 2 by each term in the second binomial: \(2 \times x = 2x\) and \(2 \times 4 = 8\).
Combining these, we get:
\[x^2 + 4x + 2x + 8 = x^2 + 6x + 8\].

Since the expanded form matches the original polynomial, we confirmed that \((x + 2)(x + 4)\) is a correct factorization.