Problem 32 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$9 m^{2}-12 m+4$$

Step-by-Step Solution

Answer

(3m - 2)^2

Step 1: Identify the form of a perfect square trinomial

A perfect square trinomial is of the form \[a^{2} - 2ab + b^{2} = (a-b)^{2} \text{ or } a^{2} + 2ab + b^{2} = (a+b)^{2}\]It factors into a binomial squared form.

Step 2: Check for a perfect square on both ends

The first term \(9m^{2}\) is \((3m)^{2}\) and the last term \(4\) is \(2^{2}\). This suggests the trinomial is a perfect square.

Step 3: Determine the binomial expression

According to the form \[a^2 - 2ab + b^2 = (a-b)^2\] Our middle term should be \(-2ab\). Set up the equation:\[-2(3m)(2) = -12m\] This matches the middle term.

Step 4: Write the binomial squared

Since \(9m^2 - 12m + 4\) is a perfect square trinomial, it can be written as \[(3m - 2)^2.\]

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special algebraic expressions. They resemble the form \[a^{2} \pm 2ab + b^{2}\]. Here's why the name makes sense:

• Square: The ending terms, \(a^2\) and \(b^2\), are squares.
• Perfect: They rearrange perfectly into a square binomial when factored.

To factor a perfect square trinomial, notice the first and last terms might be perfect squares. Like in our example, \(9m^2 - 12m + 4\):

• The first term, \(9m^2\), equals \((3m)^2\).
• The last term, \(4\), equals \(2^2\).

Together, they create the trinomial \((3m - 2)^2\). The middle term confirms this since it equals \(-2ab\), proving it factors into a binomial expression. Grasping perfect square trinomials helps streamline factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra and appear in multiple forms, like polynomials. These expressions can be simple or complex. In our example, the expression \(9m^2 - 12m + 4\) showcases variables and numbers:

• Variables: Symbols, like \(m\), represent numbers.
• Numbers: Constants, like \(9, -12,\) and \(4\), provide specific values.

Operations, such as addition, subtraction, multiplication, and division, connect these elements. By understanding expressions, you determine how to manipulate and simplify them,
as in perfect square trinomials. Recognizing patterns helps in manipulative techniques like factoring,
distribution, or combining like terms.

Binomial Expressions

Binomial expressions consist of two terms connected by an addition or subtraction sign. These expressions are simplified forms of polynomials and integral to factoring. For instance, \((a - b)^2\) is a squared binomial expression.

• Two Terms: Binomials contain exactly two elements.
• Simplification: They often simplify complex algebraic forms, like trinomials.

In the exercise's final step, you transform the trinomial \(9m^2 - 12m + 4\) into the binomial squared form \((3m - 2)^2\). Identifying binomials helps in processes like factoring and expansion. These simpler forms make algebraic manipulations more approachable,
benefiting from structure and predictability. Mastery over binomial expressions aids in better understanding algebraic principles and solving equations efficiently.