Problem 33 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 225 y^{2}+120 y+16 $$

Step-by-Step Solution

Answer

The polynomial factors as \((15y + 4)^2\).

Step 1: Identify the Common Factor

First, let's check if there is a greatest common factor (GCF) for the terms in the polynomial. The polynomial is \( 225y^2 + 120y + 16 \). We can see that there is no common factor for the coefficients 225, 120, and 16, so we're left with the polynomial as it is.

Step 2: Check the Form for a Perfect Square Trinomial

This polynomial can be viewed in the form \( ax^2 + bx + c \). To see if it's a perfect square trinomial, we notice that for \( ay^2 + by + c \), we need to have \( a = k^2 \), \( c = m^2 \), and \( b = 2km \). Here, \( a = 225 = 15^2 \), \( c = 16 = 4^2 \), and indeed, \( b = 120 = 2 \times 15 \times 4 \). Thus, this is a perfect square trinomial.

Step 3: Write the Polynomial as a Square

Since we have established that the polynomial is a perfect square, we can write it as \((15y + 4)^2\). This means the polynomial \(225y^2 + 120y + 16\) factors into the square of \(15y + 4\).

Key Concepts

Perfect Square Trinomials

Understanding perfect square trinomials can simplify the process of factoring polynomials significantly. A perfect square trinomial is a specific type of polynomial that takes the form: \( (py + q)^2 = p^2y^2 + 2pqy + q^2 \). These trinomials arise when a binomial is multiplied by itself.
To determine if a polynomial is a perfect square trinomial:

• Check if the first and last terms are perfect squares. In our example, \(225y^2\) and \(16\) are perfect squares because \(225y^2 = (15y)^2\) and \(16 = 4^2\).
• Verify that the middle term matches \(2pq\), where \(p\) and \(q\) are the square roots of the first and last terms, respectively. For \(120y\), we see that \(120 = 2 \times 15 \times 4\), confirming it fits the pattern.

Recognizing this pattern lets us directly factor the trinomial into \((15y + 4)^2\), illustrating the ease and efficiency of spotting perfect square trinomials in polynomials.

Greatest Common Factor

Finding the greatest common factor (GCF) is an essential step in factoring polynomials. The GCF of polynomial terms is the largest number and variable power that all terms share. For instance, if a polynomial’s terms are divisible by a certain number or variable, they can be factored out to simplify the expression.
Unfortunately, not all polynomials will have a GCF greater than 1, as in the polynomial \(225y^2 + 120y + 16\). Here's how you determine if there is a GCF:

• Examine the coefficients. Check for common factors across the constants in all terms. In our example, 225, 120, and 16 have no common factor other than 1.
• Consider the variables. Ensure that each term’s variable part shares a common power.

When there isn’t a GCF other than 1, as with \(225y^2 + 120y + 16\), you move to other factoring methods like recognizing perfect square trinomials.

Polynomial Expressions

Polynomial expressions consist of terms made up of variables raised to powers and multiplied by coefficients. The simplest form is a monomial, while complex ones include binomials and trinomials. Handling polynomial expressions efficiently involves recognizing patterns and applying specific factoring techniques.
Key aspects of working with polynomial expressions include:

• Degrees of terms: The degree is determined by the highest exponent on the variable(s).
• Coefficients: Numbers multiply the variable parts and determine the scale of effect for each term.

Factoring involves breaking down complex polynomials into simpler factors. This can mean identifying the greatest common factor, spotting special forms like perfect square trinomials, or using methods like grouping or the quadratic formula. In our exercise, recognizing that \(225y^2 + 120y + 16\) forms a perfect square trinomial allowed us to factor it elegantly into \((15y + 4)^2\), illustrating a core skill in polynomial manipulation.