Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 16 x^{2}+24 x+9 $$

Short answer

\((16x^2 + 24x + 9) = (4x + 3)^2\)

Step-by-step solution

Recognize the Type of Polynomial

The given polynomial is a quadratic trinomial of the form: \(a x^2 + bx + c\) Here: \(a=16\), \(b=24\), and \(c=9\).

Check for Perfect Square Trinomial

We need to see if the polynomial is a perfect square trinomial, which means it can be written in the form \((ax + b)^2\). For a perfect square trinomial, \(a\) and \(c\) should be perfect squares, and \(b\) should be twice the product of the square roots of \(a\) and \(c\).

Find Square Roots

Determine the square roots: \(\text{Square root of } 16x^2 \text{ is } 4x\) \(\text{Square root of } 9 \text{ is } 3\)

Verify Middle Term

Check if the middle term matches \(2 \times (4x) \times 3\): \(2 \times 4x \times 3 = 24x\) The middle term is \(24x\), so the polynomial is indeed a perfect square trinomial.

Write as Perfect Square

Since it is a perfect square trinomial, we factor it as: \((4x + 3)^2\)

Key concepts

Quadratic Trinomials

Quadratic trinomials are polynomials of degree 2, consisting of three terms. A typical quadratic trinomial looks like this: \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These types of polynomials are common in algebra and can be factored in several ways. The goal of factoring is to express the polynomial as a product of simpler polynomials.

Recognizing a quadratic trinomial is crucial because it helps us determine which factoring method to use. For example, the polynomial given in the exercise is \(16x^2 + 24x + 9\), where \(a = 16\), \(b = 24\), and \(c = 9\).

When working with quadratic trinomials, always first check if it can be factored easily using special techniques like perfect squares or simple factor pairs. This makes the process more straightforward and less error-prone.

Perfect Square Trinomials

Perfect square trinomials are a special type of quadratic trinomial where the polynomial is the square of a binomial. This means it can be written in the form \((ax + b)^2\). To check if a quadratic trinomial is a perfect square, follow these steps:

Identify the square roots of the first term \(a^2\) and the last term \(c\). In our example, we recognize the polynomial \(16x^2 + 24x + 9\).

• Square root of \(16x^2\) is \(4x\)
• Square root of \(9\) is \(3\)

Next, verify the middle term \(b\). The middle term in a perfect square trinomial should be twice the product of the square roots of the first and last terms. Here, \(2 \times 4x \times 3 = 24x\), which matches our middle term exactly.

This confirms that \(16x^2 + 24x + 9\) is a perfect square trinomial, and it can be written as \((4x + 3)^2\). This factorization simplifies the polynomial and makes it easier to work with in other calculations or problem-solving.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. This is an important skill in algebra and helps solve equations, simplify expressions, and identify polynomial properties.

To factor a polynomial like \(16x^2 + 24x + 9\), follow these steps:

• Identify the polynomial type: Here, it's a quadratic trinomial.
• Check for special forms such as perfect square trinomials.
• Apply appropriate factoring methods based on the polynomial type.

In this case, we found that our polynomial is a perfect square trinomial. We wrote it as \((4x + 3)^2\).

If a polynomial doesn't fit special forms, other techniques like splitting the middle term, using the quadratic formula, or synthetic division can be used. Understanding different factorization methods makes algebra problems easier to manage. Practice regularly with various types of polynomials to become proficient in this essential skill.