Question

Factor each polynomial. $$ x^{2}-10 x+16 $$

Short answer

(x - 2)(x - 8)

Step-by-step solution

Identify the polynomial

The given polynomial is \(x^{2} - 10x + 16\). This is a quadratic polynomial of the form \(ax^2 + bx + c\).

Find pairs of factors of the constant term

Look for pairs of factors of the constant term (16) that add up to the coefficient of the linear term (-10). The factors of 16 are (1, 16), (2, 8), and (4, 4). The appropriate pair that adds up to -10 is (-2) and (-8).

Write the factors in binomial form

Using the factors from Step 2, write the polynomial as a product of binomials: \((x - 2)(x - 8)\).

Verify the factorization

Expand \((x - 2)(x - 8)\) to verify: \[ (x - 2)(x - 8) = x^2 - 8x - 2x + 16 = x^2 - 10x + 16 \]. The factorization is correct.

Key concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It has the general form: \[ ax^2 + bx + c = 0 \] where:

• \(a\) is the coefficient of the quadratic term \(x^2\)
• \(b\) is the coefficient of the linear term \(x\)
• \(c\) is the constant term

This type of equation forms a parabola when graphed. Factoring is one method to solve quadratic equations. It involves expressing the quadratic polynomial as a product of two binomials. For this equation, finding two numbers that multiply to the constant term and add to the linear coefficient is key. Recognizing a quadratic form simplifies factoring and solving.

Binomial Factors

A binomial is a polynomial with exactly two terms. To factor a quadratic equation into binomial factors, you need to identify pairs of numbers that multiply to give the constant term, and add to give the middle term. For example, in the expression: \[ x^2 - 10x + 16 \] We look for pairs of numbers that:

• Multiply to 16 (the constant term)
• Add up to -10 (the coefficient of the middle term)

Here, the pair (-2) and (-8) fits because: \( (-2) \times (-8) = 16 \) and \( (-2) + (-8) = -10 \) So the binomial factors are \( (x - 2) \) and \( (x - 8) \). This gives: \[ (x - 2)(x - 8) \]

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial. The goal is to simplify the expression or to make solving equations easier. The given quadratic polynomial: \[ x^2 - 10x + 16 \] can be factored by:

• Identifying two numbers that multiply to the constant term (16) and add to the linear coefficient (-10)
• Writing the polynomial as a product of binomials using these numbers

Thus, the factorization: \[ (x - 2)(x - 8) \] simplifies to \[ x^2 - 10x + 16 \] when expanded. Factoring is like reverse multiplication. It is an essential skill in algebra that makes solving higher-degree equations easier and provides deeper insights into the polynomial's roots.