Problem 46 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ c^{2}-22 c+121 $$

Step-by-Step Solution

Answer

The factored form of the expression \(c^{2}-22 c+121\) is \((c - 11)^2\).

Step 1: Identify the 'a', 'b', and 'c' values

In the given quadratic trinomial \(c^{2}-22 c+ 121\), 'a' is 1 (the coefficient of \(c^2\)), 'b' is -22 (the coefficient of c), and 'c' is 121 (the constant term).

Step 2: Find two numbers that multiply to 'c' and add to 'b'

We need to find two numbers that multiply to 121 and add to -22. Here, -11 and -11 are the two numbers that satisfy this condition, because \(-11 \times -11 = 121\) and \(-11 - 11 = -22\).

Step 3: Factor the trinomial

We can now factor the trinomial into two binomials. Replace -22c with -11c and -11c. Thus, the factored form of the given trinomial will be \((c - 11)^2\). By the format, \((x - a)(x - b)\), when a and b are both -11.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are a specific type of algebraic expression that are very common in mathematics. They take the form of \(ax^2 + bx + c\), where:

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

In our exercise, the expression \(c^{2} - 22c + 121\) is a perfect example of a quadratic trinomial. Here, \(a = 1\), \(b = -22\), and \(c = 121\).
Quadratic trinomials are often encountered when solving polynomial equations, graphing parabolas, or factoring expressions. Understanding the structure of these trinomials makes it easier to simplify them by finding their factors. The challenge usually lies in learning how the coefficients and constant term interact with each other to transform the quadratic trinomial into a product of binomials.

Binomial Factoring

Binomial factoring involves expressing a trinomial as a product of two binomial expressions. This is an essential skill in algebra, especially when working with quadratic trinomials.
The method seeks numbers that satisfy two conditions:

• Their sum should equal the middle coefficient (\(b\))
• Their product should equal the constant term (\(c\))

From our problem, for \(c^{2} - 22c + 121\), numbers \(-11\) and \(-11\) multiply to \(121\) and add up to \(-22\). These numbers were used to split the middle term and factor the trinomial into \((c - 11)(c - 11)\) or written simply as \((c - 11)^2\).
Factoring into binomials is powerful because it simplifies complex equations and prepares them for solving. By factoring, we can easily find x-intercepts or any roots a quadratic equation might have.

Identifying Coefficients

Identifying coefficients is the first crucial step in working with quadratic expressions and plays a significant role in successful factoring or graphing.
In any quadratic trinomial \(ax^2 + bx + c\):

• \(a\) tells us the parabola's direction (opens upwards if positive, downwards if negative)
• \(b\) depicts the parabola's horizontal shift
• \(c\) usually stands as the y-intercept on a graph

For the given equation \(c^{2} - 22c + 121\), we found \(a = 1\), \(b = -22\), and \(c = 121\). Recognizing these coefficients allows us to rearrange and manipulate the expression confidently.
Understanding these coefficients helps in predicting the behavior of the parabola and tackling various mathematical problems such as finding the vertex, axis of symmetry, and roots.