Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Factor the perfect square trinomial.\(x^{2}+10 x+25\)

Step-by-Step Solution

Answer

The factored form of \(x^{2}+10x+25\) is \((x+5)^{2}\).

Step 1: Identify the pattern

Observe that \(x^{2}+10x+25\) fits the pattern of a perfect square trinomial, \(a^{2} + 2ab + b^{2}\).

Step 2: Validate the pattern

Make sure that the trinomial complies with the pattern of a perfect square trinomial. In this case, \(a = x\), \(b = 5\), and hence the middle term is \(2 * x * 5 = 10x\), which validates our pattern.

Step 3: Factor the trinomial

Now that we've confirmed that the trinomial fits the pattern of a perfect square trinomial, it can be factored into the square of binomial. Therefore, \(x^{2}+10x+25\) can be factored into \((x+5)^{2}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, consisting of numbers, variables, and operators grouped together to represent a value or equation. An example includes polynomial expressions like the one in our exercise: \(x^2 + 10x + 25\). Here, we see numbers and variables combined through addition and multiplication. It's important to recognize that algebraic expressions can take many forms, from simple monomials, such as \(5x\), to more complex combinations called polynomials. These polynomials, like \(ax^2 + bx + c\), can be categorized based on the number of terms they contain and their degree. The process of evaluating, simplifying, and manipulating these expressions, including factoring, is vital for solving algebraic problems. In our exercise, recognizing the expression as a perfect square trinomial simplifies our task. Understanding algebraic expressions is the first step towards solving equations and inequalities.

Trinomial Patterns

Trinomial patterns are essential in determining the nature of an algebraic expression. A trinomial is a polynomial with three terms, such as \(ax^2 + bx + c\). Identifying specific patterns in trinomials can make factoring more intuitive.

In our exercise, \(x^2 + 10x + 25\) fits the pattern of a perfect square trinomial. This type of trinomial follows the form \(a^2 + 2ab + b^2\), where \(a\) and \(b\) are terms that can be factored. The identification process requires checking the middle term, \(2ab\), to see if it results from doubling the product of the terms \(a\) and \(b\).

To identify the perfect square pattern:

• Check if the first and last terms are perfect squares.
• Confirm that the middle term is twice the product of the square roots of the first and last terms.

This understanding assists greatly in recognizing that \(x^2 + 10x + 25\) is indeed a perfect square trinomial, allowing it to be factored efficiently.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler "factorable" components, or factors, which when multiplied together give the original polynomial. Recognizing patterns in polynomials is crucial for effective factoring.

In perfect square trinomials, we can simplify the trinomial by factoring it into a binomial squared. In our exercise, after confirming the pattern \(a^2 + 2ab + b^2\), we notice that \(a = x\) and \(b = 5\). Hence, \(x^2 + 10x + 25\) factors into \((x + 5)^2\).

Here are key steps to polynomial factoring:

• Identify if the polynomial fits a known pattern (like a perfect square trinomial).
• Break it down into its component binomials.
• Verify by expanding the binomials to check if they accord with the original polynomial.

Factoring is essential in algebra as it allows you to simplify expressions and solve equations more easily by setting each factor to zero.