Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{2}-6 x+8 $$

Short answer

(x - 2)(x - 4)

Step-by-step solution

Identify the polynomial

Given the polynomial Given the polynomial is x^2 - 6x + 8. The polynomial is a quadratic expression of the form ax^2 + bx + c where a = 1, b = -6, c = 8.

Find the factors of the constant term

Consider the constant term which is 8. Identify pairs of factors of 8: The factors of 8 are (1, 8), (2, 4), (-1, -8), and (-2, -4).

Identify the factor pair that sums to the middle term

Since the middle term is -6, determine which pair of factors adds up to -6: Out of the factor pairs (1, 8), (2, 4), (-1, -8), and (-2, -4), only -2 and -4 add up to -6.

Write the factorized form

Rewrite the polynomial using the identified factor pair: The polynomial can be factored as (x - 2)(x - 4). Verify by expanding (x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8.

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \[ ax^2 + bx + c \] where a, b, and c are constants, and x is the variable. They are known as 'quadratic' because the term 'quad' means square, and the variable x is squared \( (x^2) \). These equations often appear in various mathematical and real-world problems. To solve or simplify quadratic equations, understanding the concepts of factoring is crucial. Factoring involves breaking down a complex expression into simpler components or factors. When you can write a quadratic equation as a product of two binomials, solving the equation becomes straightforward. In the given example \( x^2 - 6x + 8 \), we need to rewrite the quadratic equation as a product of its factors. This helps in identifying the values of x that satisfy the equation (making the equation equal to zero).

Factor Pairs

To factor a quadratic equation, such as \( x^2 - 6x + 8 \), we need to find the factor pairs of the constant term 'c'. The constant term here is 8, so we look for pairs of numbers that multiply to give 8. These pairs are:

• (1, 8)
• (2, 4)
• (-1, -8)
• (-2, -4)

The next step involves identifying which of these pairs adds up to the coefficient of the middle term, 'b'. In this example, the coefficient of 'b' is -6. We need to find the pair that sums to -6. By examining our factor pairs, we see that \( -2 + (-4) = -6 \). Therefore, \( (-2, -4) \) is the correct factor pair for the polynomial. This process allows us to decompose and simplify our quadratic equation effectively.

Factored Form

Factoring a polynomial involves writing it as a product of simpler expressions, or factors. For quadratic equations, this typically means expressing them as the product of two binomials. In our example \( x^2 - 6x + 8 \), we identified \( (-2, -4) \) as the factor pair. Using these factors, we rewrite the polynomial: \[ (x - 2)(x - 4) \] To verify our work, we can expand the factored form and check if it matches the original polynomial: \[ (x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8 \] This shows that the factored form is correct. Factoring quadratics into their factored form makes solving equations simpler because it transforms a complex polynomial into a product of simpler binomials, making the roots of the equation easy to find.