Problem 93 - Algebra A Combined Function

Question

Factor each trinomial completely. \(4 x^{2}(y-1)^{2}+25 x(y-1)^{2}+25(y-1)^{2}\)

Step-by-Step Solution

Answer

The trinomial factors to \((y-1)^{2}(2x + 5)^{2}\).

Step 1: Identify Common Factor

The given trinomial is \(4x^{2}(y-1)^{2} + 25x(y-1)^{2} + 25(y-1)^{2}\). Observe that \((y-1)^{2}\) is a common factor for each term.

Step 2: Factor Out the Common Factor

Extract \((y-1)^{2}\) from the trinomial, which gives: \((y-1)^{2}(4x^{2} + 25x + 25)\).

Step 3: Recognize the Perfect Square Trinomial

Identify that the expression inside the parentheses, \(4x^{2} + 25x + 25\), is a perfect square trinomial. It can be rearranged into the form \((2x + 5)^{2}\) since: - \((2x)^{2} = 4x^{2}\), - \(2 \cdot (2x) \cdot 5 = 20x + 5x = 25x\), - \(5^{2} = 25\).

Step 4: Write the Completely Factored Form

Substitute the perfect square trinomial back into the expression to write the fully factored form: \((y-1)^{2}(2x + 5)^{2}\).

Key Concepts

Common Factor

When factoring trinomials, one of the first steps is to look for a common factor. A common factor is a term that appears in every part of the expression. Identifying this can significantly simplify the process of factoring.

In the given trinomial, we see that \((y-1)^{2}\) is a common factor in all the terms: \(4x^{2}(y-1)^{2}\), \(25x(y-1)^{2}\), and \(25(y-1)^{2}\). This means that each part of the expression includes \((y-1)^{2}\), making it an ideal candidate to be factored out.

• Extracting common factors can simplify the expression.
• It sets the stage for further factoring steps.

By factoring out \((y-1)^{2}\), you simplify the trinomial, making the remaining parts easier to work with. In essence, identifying a common factor can be seen as 'clearing the path' towards a more straightforward factoring job. After this step, any further operations need only be performed on the simplified version of the trinomial.

Perfect Square Trinomial

A perfect square trinomial is a special pattern which can be factored into a binomial squared. Recognizing these patterns can greatly reduce the complexity of factoring.

In this exercise, once \((y-1)^{2}\) is factored out, we are left with the expression \(4x^{2} + 25x + 25\). This expression is a classic example of a perfect square trinomial.

• First, notice that \((2x)^{2} = 4x^{2}\).
• Then we check if the middle term, \(25x\), matches the pattern \(2 imes ext{{first term}} imes ext{{last term}}\).
• Finally, verify that \(5^{2} = 25\) is indeed the last term.

This trinomial matches the pattern because it can be rewritten in the form \((2x + 5)^{2}\). Understanding this special type of trinomial allows you to skip several factoring steps and directly write the expression in its factored form. Recognizing perfect square trinomials saves time and ensures accuracy.

Factoring Techniques

Factoring techniques are the fundamental methods used to break down complex polynomials into simpler expressions.

In the given trinomial, a combination of techniques is applied:

• Initial factoring involves finding and removing common factors, in this case, \((y-1)^{2}\), to simplify the trinomial.
• Next, recognizing the structure of a perfect square trinomial helps us use the technique more efficiently.
• Finally, applying these combined techniques results in the completely factored form: \((y-1)^{2}(2x + 5)^{2}\).

These techniques include systematic steps like identifying common factors and using special formulas such as perfect square trinomials. The goal of these techniques is to "break down" polynomial expressions into simpler, interpretable forms. Mastering such techniques will equip you to handle increasingly complex polynomial forms, making algebra not just about solving equations, but understanding their core structure and form.