Problem 62 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 9 x^{2}+24 x+16 $$

Step-by-Step Solution

Answer

The polynomial factors to \((3x + 4)^2\).

Step 1: Identify the Form

Notice that the given quadratic polynomial is of the form \( ax^2 + bx + c \), where \( a = 9 \), \( b = 24 \), and \( c = 16 \).

Step 2: Recognize a Perfect Square

Determine if the polynomial is a perfect square trinomial. A perfect square trinomial usually follows the pattern \((ax+b)^2 = a^2x^2 + 2abx + b^2\). Identify if this structure fits your polynomial.

Step 3: Find the Square Roots

Check if \( 9x^2 \) and \( 16 \) are perfect squares: \( 9x^2 = (3x)^2 \) and \( 16 = 4^2 \). Both are perfect squares.

Step 4: Check Middle Term

Verify the middle term \( 24x \) corresponds to \( 2abx \), where \( a = 3x \) and \( b = 4 \). This gives \( 2 \times 3x \times 4 = 24x \), which matches the middle term in the polynomial.

Step 5: Write as a Perfect Square

Since the terms fit the pattern of a perfect square trinomial, the polynomial can be written as \((3x + 4)^2\).

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, which means its highest exponent is 2. Typically, you will see it in the form \( ax^2 + bx + c \). In this form:

• \(a\), \(b\), and \(c\) represent constants
• \(x\) is the variable
• \(ax^2\) indicates the quadratic term
• \(bx\) represents the linear term
• \(c\) is the constant term

Quadratic expressions are fundamental in algebra, often used to describe parabolic graphs when plotted. When dealing with quadratics, identifying the terms \(a\), \(b\), and \(c\) helps in processes like factoring or finding the roots of the equation. Understanding these basics is essential when you proceed to factor quadratic expressions or convert them into different forms.

Perfect Square Trinomials

A perfect square trinomial takes the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). Recognizing this form is crucial because it simplifies the process of factoring. When you have a trinomial:

• Check if the first term is a perfect square, like \((3x)^2 = 9x^2\)
• Ensure the last term, \(c\), is a perfect square, such as \(4^2 = 16\)
• Confirm that the middle term, \(b\), matches \(2abx\); for example, here, \(24x = 2 \times 3x \times 4\)

When all these conditions are met, the trinomial is a perfect square. It can then be rewritten, greatly simplifying solving or graphing tasks. Recognizing patterns like these distinguishes perfect square trinomials from other quadratic equations.

Polynomial Factorization

Polynomial factorization is a key skill in algebra that involves breaking down a polynomial into products of simpler polynomials or binomials. For quadratics, this means expressing it in the form \((px + q)(rx + s)\). The benefits of factoring include:

• Making complex expressions simpler to handle
• Finding the roots of polynomials easily
• Aiding in solving equations efficiently

Perfect square trinomials are among the simplest to factor because they follow a predictable pattern. When you confirm a quadratic as a perfect square trinomial, you can immediately write it in its factored form, such as \((3x + 4)^2\) in this case. Factoring transforms the polynomial into its most useful form for further algebraic manipulation.

Simplifying Expressions

Simplifying expressions is essential for making mathematical problems easier to solve. This process involves reducing an expression to its simplest form by combining like terms, removing parentheses, and performing basic arithmetic operations. Simplification helps us:

• Make sense of complex problems quickly
• Reveal underlying structures, such as squares or roots
• Prepare the expression for solving or graphing

For example, factoring a complex trinomial as a perfect square, and expressing it as \((3x + 4)^2\), simplifies further algebraic work and helps understand the geometric representation. Simplification not only aids in problem-solving efficiency but also enhances comprehension of algebraic concepts.