Problem 13 - Beginning and Intermediate Algebra

Question

Factor completely. $$4 y^{2}+12 y+9$$

Step-by-Step Solution

Answer

The completely factored form of the expression \(4y^2 + 12y + 9\) is \((2y + 3)^2\).

Step 1: Identify common factors in the coefficients

First, let's find the common factors in the coefficients of the given expression \(4y^2 + 12y + 9\). The coefficients are 4, 12, and 9. Their greatest common divisor (GCD) is 1, which means there is no common factor to factor out.

Step 2: Observe the quadratic expression

The given expression is of the form \(ay^2 + by + c\), where a = 4, b = 12, and c = 9. Notice that these coefficients are all positive and that the expression is a perfect square trinomial. Therefore, we can use the following formula to factor the expression: \((py + q)^2 = p^2y^2 + 2pqy + q^2\).

Step 3: Find the values of p, q

We need to find the values of p and q, such that \(a = p^2\), \(c = q^2\), and \(b = 2pq\). Given that a = 4, b = 12, and c = 9, we can let: - p = 2, as \(p^2 = 2^2 = 4\) - q = 3, as \(q^2 = 3^2 = 9\) So indeed we have: $$b = 2pq = 2(2)(3) = 12.$$ So, the given expression can be written as \((2y + 3)^2\).

Step 4: Write down the completely factored expression

The final factored form of the given expression \(4y^2 + 12y + 9\) is \((2y + 3)^2\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic expression. These trinomials can be factored into a binomial squared. The standard form of a perfect square trinomial is written as:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

When you recognize a quadratic in this format, you can quickly factor it as a perfect square. This makes solving for its roots or simplifying expressions much easier.
In our expression \(4y^2 + 12y + 9\), we identify it as a perfect square trinomial by noting:

• The first term \(4y^2\) is \((2y)^2\).
• The last term \(9\) is \(3^2\).
• The middle term \(12y\) fits the format \(2 \cdot 2y \cdot 3\).

Thus, the expression simplifies to the square of a binomial, \((2y + 3)^2\). Identifying these structures helps in simplifying and solving polynomial equations with ease.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a concept used to find the largest factor that divides all terms in a polynomial without leaving a remainder. It is a useful tool when trying to simplify expressions before factoring further. To determine the GCD of polynomial coefficients, look for the largest number that divides each coefficient.
In the expression \(4y^2 + 12y + 9\), the coefficients are 4, 12, and 9. Checking these coefficients, the divisors are:

• 4: 1, 2, 4
• 12: 1, 2, 3, 4, 6, 12
• 9: 1, 3, 9

The only common factor for all is 1, meaning there isn’t a larger common factor to factor out. Recognizing a low GCD indicates the expression cannot be simplified by common factors alone and directs us to consider other factoring techniques like detecting perfect square trinomials.

Polynomial Expressions

Polynomial expressions are sums of terms consisting of variables raised to whole number powers and their coefficients. Understanding polynomials is foundational for algebra. They can range from simple

• Monomials: Single term, like \(5x^3\)
• Binomials: Sum or difference of two terms, like \(2x + 3\)
• Trinomials: Three terms, such as \(x^2 + 5x + 6\)

The expression \(4y^2 + 12y + 9\) is an example of a quadratic trinomial since it has three terms involving \(y\) with the highest exponent being 2.
Factoring polynomial expressions involves breaking them down into products of simpler polynomials. This makes equations easier to solve, particularly when determining their roots. Recognizing forms like perfect square trinomials simplifies the factoring process. Factoring relies on knowledge about things like the structure of expressions and using tools like the GCD to guide the process.