Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Factor the perfect square trinomial.\(9 x^{2}-12 x+4\)

Step-by-Step Solution

Answer

The factor of the perfect square trinomial \(9 x^{2}-12 x+4\) is \((3x - 2)^2\)

Step 1: Identify the Trinomial

The given trinomial is \(9 x^{2}-12 x+4\)

Step 2: Take the square root of the first and third terms

Take the square root of the first term \(9 x^{2}\) which is \(3x\) and of the third term \(4\), which is \(2\).

Step 3: Formulate the perfect square trinomial

Now the square of binomial can be formed i.e., \((3x - 2)^2\). Verify by expanding the binomial square and compare the result with the original trinomial.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of trinomial where it can be expressed as the square of a binomial. This means that when you factor the trinomial, you end up with

• The same value appearing twice as a repeated binomial.

In mathematical terms, a perfect square trinomial has the form \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\).
For example, the trinomial \(9x^2 - 12x + 4\) is a perfect square trinomial. Here, when you factor it, the result is \((3x - 2)^2\). You can see this by taking the square root of the first term \(9x^2\) resulting in \(3x\), and the square root of the last term \(4\) resulting in \(2\).
The middle term \(-12x\) is

• The product of the two roots \(3x\) and \(-2\), multiplied by 2, which confirms the perfect square trinomial pattern.

Binomial Expansion

The process of expanding a binomial involves multiplying the binomial by itself to get a trinomial. This is known as binomial expansion. When you have a squared binomial like \((a + b)^2\), binomial expansion helps us rewrite it as \(a^2 + 2ab + b^2\).
Here’s how it works: multiply each term in the binomial by every term in the other binomial:

• First, multiply \(a\) by \(a\) and \(b\).
• Then, multiply \(b\) by \(a\) and \(b\).

This will unfold into a trinomial where

• The first term is \(a^2\),
• The middle term is \(2ab\),
• And the last term is \(b^2\).

For instance, if you expand \((3x - 2)^2\), it results in \(9x^2 - 12x + 4\), verifying the perfect square trinomial.

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree 2, typically in the form \(ax^2 + bx + c\). A perfect square trinomial is a type of quadratic expression.
This specific structure suggests that the expression can be squared to achieve or revert to a simpler binomial.
Understanding quadratic expressions involves recognizing:

• The coefficient of \(x^2\) (the leading term that determines the form of expansion),
• The linear coefficient (the term affecting symmetry of the graph if plotted),
• The constant term (that shifts the entire graph up or down).

In our example \(9x^2 - 12x + 4\),

• 9 is the leading coefficient indicating perfect squares are part of the expression,
• -12x reflects the type and direction of linear shifts,
• while 4 represents vertical translation.

Recognizing these properties is essential in factoring and solving equations related to quadratics. This lays the foundation for efficiently working with deeper algebra concepts.