Problem 33 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The factored form is \((15y + 4)^2\).

Step 1: Identify Type of Polynomial

The polynomial given is \(225y^2 + 120y + 16\). It is a quadratic polynomial in the standard form \(ay^2 + by + c\), where \(a = 225\), \(b = 120\), and \(c = 16\).

Step 2: Check for Perfect Square Trinomial

A perfect square trinomial takes the form \((my + n)^2 = m^2y^2 + 2mny + n^2\). We compare this with the given polynomial. Here, \(m^2 = 225\), so \(m = 15\), and \(n^2 = 16\), so \(n = 4\). Then, calculate \(2mn = 2 \times 15 \times 4 = 120\), which matches \(b\) in the polynomial.

Step 3: Write as a Square of Binomial

Since the terms match perfectly, the given polynomial can be expressed as the square of a binomial. Thus, \(225y^2 + 120y + 16 = (15y + 4)^2\).

Step 4: Verify the Factorization

To verify, expand \((15y + 4)^2\) to ensure it returns to the original polynomial. \((15y + 4)^2 = 15^2y^2 + 2 \times 15 \times 4y + 4^2 = 225y^2 + 120y + 16\). The factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It is the square of a binomial. You can often spot these trinomials as they have a neat structure:

• They take the form \(a^2x^2 + 2abx + b^2\), which resembles \((ax + b)^2\).
• To recognize a perfect square trinomial, consider checking if both the first and last terms are perfect squares.
• Then, confirm that the middle term fits in the pattern 2 times the product of the square roots of the first and last terms.

In the example polynomial \(225y^2 + 120y + 16\), the terms are:

• \(225y^2\) is \(15^2y^2\).
• \(16\) is \(4^2\).
• The middle term \(120y\) matches \(2 \times 15 \times 4y\).

All conditions for a perfect square trinomial are satisfied. Therefore, it confirms that \(225y^2 + 120y + 16 = (15y + 4)^2\).

Binomial

A binomial is a polynomial with exactly two terms. Binomials are fairly simple in structure but can lead to various forms depending on their usage. When dealing with perfect square trinomials, the relationships between the binomial and the trinomial are:

• The trinomial represents the expanded form of a squared binomial.
• In this context, the binomial \(ax + b\) is the root structure behind the perfect square trinomial.

For our given exercise, once you identify the structure of a perfect square trinomial, you can easily find the binomial which, when squared, gives you the trinomial. In \(225y^2 + 120y + 16\), the binomial is \(15y + 4\). Repackaging this trinomial as \((15y + 4)^2\) shows us the simple binomial that underlies its square.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. These factors are polynomials of lower degrees which multiply to give the original polynomial. An important type of factorization involves recognizing and using special patterns like perfect square trinomials, which allow for quick simplification.

• When dealing with quadratic polynomials, the goal is often to express them as a product of binomials.
• In this case, using the pattern recognition of a perfect square trinomial allows us to factorize the polynomial directly.
• Solving the polynomial \(225y^2 + 120y + 16\) into \((15y + 4)^2\) reduces its complexity and shows its roots directly.

Factorizing polynomials simplifies solving equations, finding roots, and understanding the polynomial's behavior. Recognizing patterns such as perfect square trinomials makes factorization straightforward and powerful, especially in learning and solving algebraic problems.