Problem 52 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}+24 x+144$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(x^2 + 24x + 144\) is \((x+12)^2\).

Step 1: Identify the Coefficients

The given polynomial \(x^2 + 24x + 144\) has coefficients a=1, b=24 and c=144 as per the standard quadratic equation \(ax^2 + bx + c\).

Step 2: Check for Perfect Square Trinomial

To check if the trinomial is a perfect square, we check if the square root of the first coefficient 'a' and the last coefficient 'c' are equal to half of the middle coefficient 'b'. So we express b = \(2*sqrt(a*c)\). The middle coefficient of the trinomial is 24, which can be expressed as \(2*sqrt(1*144)\), confirming the trinomial is a perfect square.

Step 3: Factor the Polynomial

We can now write the given trinomial as \(x^2 + 2*12x + 12^2\), which is in the structure \((ax+b)^2 = a^2x^2+2abx+b^2\). With this, we can simplify the trinomial into the perfect square binomial form \((x+12)^2\).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic expressions that take the form \( (ax+b)^2 \) or \( (ax-b)^2 \). These trinomials can be expanded into \( a^2x^2 \pm 2abx + b^2 \). To determine if a trinomial is a perfect square, you'll want to look for these characteristics:

• The first and last terms are perfect squares themselves.
• The middle term can be expressed as twice the product of the square roots of the first and last terms.

For example, in the polynomial \( x^2 + 24x + 144 \), both \( x^2 \) and \( 144 \) are perfect squares. The square root of \( 144 \) is \( 12 \), and the middle term \( 24x \) is twice the product of \( x \) and \( 12 \) (\( 2 \times x \times 12 = 24x \)). This confirms that \( x^2 + 24x + 144 \) is a perfect square trinomial.

Quadratic Equations

Quadratic equations are algebraic expressions that involve a variable raised to the power of 2. They are usually written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Solving these equations often involves factoring the quadratic expression, completing the square, or using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Factoring is often the simplest method, particularly when the quadratic expression is a perfect square trinomial. In our example of \( x^2 + 24x + 144 \), recognizing it as a perfect square trinomial allows us to factor it as \( (x+12)^2 \), which simplifies the solving process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can take many forms, such as polynomials, rational expressions, or radical expressions. Understanding how to manipulate and simplify these expressions is key to solving algebraic problems.
For instance, a polynomial is a type of algebraic expression that includes terms such as \( ax^n \), where \( n \) is a non-negative integer. In our context, when we factor expressions like \( x^2 + 24x + 144 \), we are transforming them into simpler or more useful forms, such as the perfect square binomial \( (x+12)^2 \).
This transformation process involves breaking down complex expressions into components that are easier to handle, offering clearer insights into their structure and potential solutions.