Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 4 x^{2}+12 x+9 $$

Short answer

(2x + 3)^2

Step-by-step solution

- Identify the polynomial

First, identify the given polynomial. The polynomial given is 4x^2+12x+9

- Check if it is a perfect square trinomial

Check if the polynomial is a perfect square trinomial. A perfect square trinomial has the form a^2 + 2ab + b^2. Here, 4x^2 can be written as (2x)^2, 12x as 2(2x)(3), and 9 as 3^2. Indeed, 4x^2 + 12x + 9 = (2x)^2 + 2(2x)(3) + 3^2

- Factor as a binomial squared

Since we have identified it as a perfect square trinomial, factor it as a binomial squared. The polynomial can be factored as follows : (2x + 3)^2

Key concepts

Perfect square trinomial

A perfect square trinomial is a special type of polynomial. It follows the form \( a^2 + 2ab + b^2 \). This is useful because it can be factored into \( (a + b)^2 \). These trinomials are called 'perfect squares' because they come from squaring a binomial.

Consider our given polynomial: \( 4x^2 + 12x + 9 \).

• The term \( 4x^2 \) is a square itself: \( (2x)^2 \).
• The constant term \( 9 \) is also a square: \( 3^2 \).
• The middle term \( 12x \) is twice the product of \( 2x \) and \( 3 \), making it \( 2(2x)(3) \).

This meets the criteria for a perfect square trinomial. So \( 4x^2 + 12x + 9 \) can be factored into \( (2x + 3)^2 \).

Binomial factoring

Binomial factoring is a technique used to factor polynomials into products of binomials. For perfect square trinomials, this process is straightforward. After identifying the polynomial as a perfect square trinomial, it can be rewritten as the square of a binomial.

In our example, we already identified \( 4x^2 + 12x + 9 \) as \( (2x)^2 + 2(2x)(3) + 3^2 \).
To factor it, we write it as \( (2x + 3)^2 \). This means that we have found our binomial: \( (2x + 3) \).

Therefore, factoring using the methods of binomial and perfect squares simplifies to:
\( 4x^2 + 12x + 9 = (2x + 3)^2 \).
This binomial multiplication confirms that our factorization is correct.

Polynomial factoring

Polynomial factoring involves breaking down a polynomial into simpler polynomials (or factors) that multiply to produce the original polynomial. Factoring is essential for solving polynomial equations, simplifying expressions, and finding roots.

There are different techniques for factoring, such as:

• Factoring out the greatest common factor (GCF).
• Factoring by grouping.
• Factoring trinomials (like our example).
• Special products (like difference of squares and perfect square trinomials).

Our example falls under special products because identifying perfect square trinomials is a straightforward approach to polynomial factoring. The equation \( 4x^2 + 12x + 9 \right (2x + 3)^2 \) shows how recognizing special patterns in polynomials simplifies the factorization process. Once identified, it can be smoothly converted into a squared binomial.