Problem 17 - Algebra 2

Question

Factor each polynomial. \(x^{2}-7 x+6\)

Step-by-Step Solution

Answer

The factors are \((x - 6)(x - 1)\).

Step 1: Identify the Quadratic

The given polynomial is a quadratic expression in the standard form: \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -7\), and \(c = 6\).

Step 2: Find the Product and Sum

Find two numbers that multiply to \(c = 6\) and add to \(b = -7\). The numbers that satisfy this condition are \(-6\) and \(-1\) because \((-6) \times (-1) = 6\) and \((-6) + (-1) = -7\).

Step 3: Write the Binomial Factors

Express the quadratic in terms of its factors. The original expression \(x^2 - 7x + 6\) can be factored as \((x - 6)(x - 1)\) because these factors expand to give the original quadratic when multiplied together.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial that involves terms up to the second degree. In simple terms, it's an expression in the form of \(ax^2 + bx + c\), where:

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

Quadratics are widely used in algebra and are foundational for solving problems that involve parabolas.
In our example, the expression is \(x^2 - 7x + 6\). Here, \(a = 1\), \(b = -7\), and \(c = 6\). Recognizing that it's a quadratic helps us know the strategies for simplifying or solving it, such as using the quadratic formula or factoring.

Standard Form of Quadratic

The standard form of a quadratic expression makes it easier to identify the components needed for various operations. The form \(ax^2 + bx + c\) is considered the standard form. This not only helps in identifying the coefficients \(a\), \(b\), and \(c\) but also aids in applying methods like factoring and solving formulas.
In our case, having the expression \(x^2 - 7x + 6\) in the standard form means we see immediately that \(a = 1\), which simplifies the factoring process since the leading coefficient doesn’t complicate the math. It becomes straightforward to understand how to break down the quadratic into simpler expressions or factors.

• The standard form alerts us to look for two numbers that multiply to \(c\) and add to \(b\).
• This insight is critical for accurate and efficient factoring.

Binomial Factors

Factoring quadratics frequently results in two binomial factors. These are simpler expressions involving two terms each, set in parentheses, which multiply together to give the original quadratic expression. To factor a quadratic like \(x^2 - 7x + 6\), you look for two numbers that accomplish a specific task:

• They multiply to the constant term \(c\).
• They add up to the coefficient \(b\) of the linear term.

The numbers \(-6\) and \(-1\) satisfy these requirements because \((-6) \times (-1) = 6\) and \((-6) + (-1) = -7\).
Thus, the quadratic can be factored into \((x - 6)(x - 1)\). These factors are not only solutions to the expression but also provide insight into its roots, guiding us to x-values that make the expression zero. This is the essence of factoring quadratics into binomial factors.