Problem 25 - Elementary and Intermediate Algebra

Question

Factor. $$ c^{2}-12 c+36 $$

Step-by-Step Solution

Answer

The factorization of \( c^2 - 12c + 36 \) is \((c-6)^2\).

Step 1: Recognize the quadratic

The expression is a quadratic trinomial of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -12 \), and \( c = 36 \). We need to factor this expression.

Step 2: Identify perfect square trinomial

Recognize if it's a perfect square trinomial. Since both the first and last terms are perfect squares (the first, \( c^2 \), the last, \( 6^2 \)), test to see if the middle term fits the pattern \( 2ab \), where \( a = c \) and \( b = 6 \). Check: \( 2 \times c \times 6 = 12c \). Indeed, it perfectly matches \( -12c \).

Step 3: Write the trinomial as a square of a binomial

The trinomial \( c^2 - 12c + 36 \) can thus be written as \((c-6)^2\).

Step 4: Verification

To verify, expand \((c-6)^2\) to ensure it equals the original expression: \((c-6)(c-6) = c^2 - 6c - 6c + 36 = c^2 - 12c + 36\). This confirms our factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be rewritten as the square of a binomial. To identify a perfect square trinomial, we should examine the expression's structure. Typically, it takes the form \(a^2 \pm 2ab + b^2\). Here, both the first term, \(a^2\), and the last term, \(b^2\), are perfect squares.
To check if a trinomial is a perfect square, follow these steps:

• Identify the square roots of the first and last terms.
• Determine if the middle term is twice the product of these roots.
• If so, the trinomial can be factored into \((a \pm b)^2\).

Using the example of \(c^2 - 12c + 36\), we see the first term is \(c^2\) and the last term \(6^2\), both perfect squares. The middle term, -12c, matches \(-2 \times c \times 6\). Therefore, it is a perfect square trinomial, factored as \((c-6)^2\).

Binomial Expansion

Binomial expansion is the process of expanding an expression raised to a power. It is particularly useful in verifying factorizations. The expansion of a binomial squared, \((a \pm b)^2\), yields the perfect square trinomial form.
For example, consider the binomial \((c-6)^2\):

• First, square the first term: \(c^2\).
• Second, multiply the two terms and double the result: \(-2 \times c \times 6 = -12c\).
• Finally, square the second term: \(36\).

Thus, the expanded form is \(c^2 - 12c + 36\), which confirms our factorization was correct. Binomial expansion ensures that both multiplication and addition principles are followed accurately in expanding any binomial expression.

Quadratic Expression

Quadratic expressions are polynomial expressions of degree 2, generally written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The solutions or roots of these expressions can be identified either by factoring or using the quadratic formula.
Factoring is often sought after when the quadratic is a perfect square trinomial. Recognizing the form can simplify the process significantly, as seen in the exercise example with \(c^2 - 12c + 36\).
Key characteristics include:

• The highest degree is 2 (i.e., the exponent of the variable in the highest term).
• It can sometimes be factored into two binomials or a perfect square.
• Factorization helps simplify solving quadratic equations.

Understanding these concepts aids in recognizing patterns and simplifying expressions, making problems more approachable.