Problem 80 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{2}+10 x+25\)

Step-by-Step Solution

Answer

The expression factors to \((x + 5)^2\).

Step 1: Identify the quadratic expression

The given expression to factor is a quadratic trinomial: \(x^2 + 10x + 25\). We need to find two binomials that multiply to give this trinomial.

Step 2: Understand perfect square trinomials

Recognize that this expression resembles a perfect square trinomial, which takes the form \((a+b)^2 = a^2 + 2ab + b^2\). Check if it fits this form.

Step 3: Calculate square roots

Find the square root of \(x^2\), which is \(x\), and the square root of \(25\), which is \(5\). These are potential terms for the binomial factors.

Step 4: Double check middle term

To verify if the expression is a perfect square, calculate \(2ab = 2 \times x \times 5 = 10x\). Since this matches the middle term of the trinomial, it confirms that the expression is a perfect square.

Step 5: Write the binomial square

With the verification complete, express the trinomial as a squared binomial: \((x + 5)^2\). This is the factorized form of the expression.

Key Concepts

Factoring Trinomials

Factoring trinomials is an essential skill in algebra that helps simplify expressions and solve quadratic equations. Consider a trinomial in the form of \(ax^2 + bx + c\). The goal is to break it down into two simpler expressions, specifically binomials, that multiply to give the original trinomial.

• Identify the terms: The expression consists of three terms where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
• Look for factor combinations: Focus on finding two numbers that multiply to \(ac\) and add up to \(b\). These numbers will help in rewriting the trinomial in a factorable form.
• Reorganize the expression: Split the middle term using the two numbers found, and factor by grouping to find the simplest binomial factors.

By breaking down the trinomial into a pair of binomials, you can simplify the process of solving quadratic equations and finding roots.

Perfect Square Trinomials

Recognizing perfect square trinomials can speed up the factoring process significantly. A perfect square trinomial is an expression that can be rewritten as the square of a binomial.

• Form Identification: These trinomials generally take the form \((a + b)^2 = a^2 + 2ab + b^2\). Verify if your trinomial matches this pattern.
• Square Roots: Calculate the square roots of the first and the last terms. These roots offer the candidate values for the binomial.
• Double-check Middle Term: To confirm it's a perfect square trinomial, ensure that the double product of the terms from the square roots yields the middle term of the trinomial.

In our example \(x^2 + 10x + 25\), identifying it as a perfect square allows us to factor it efficiently to \((x + 5)^2\). Knowing the pattern makes solving and factoring quicker and more intuitive.

Binomial Factors

Binomial factors are the expressions that, when multiplied together, result in the original trinomial or polynomial.

• Result of Factoring: Once a trinomial is identified and verified as a perfect square, the binomial factors can be directly written as the square roots of the respective terms.
• Writing the Expression: Express the trinomial as a squared expression, for example, \((x + 5)^2\), clearly showing its binomial factors.
• Verification and Application: Always review your work to ensure the binomial multiplication returns the original trinomial. This is a role of binomial factors in understanding and solving quadratic equations.

By mastering the process of reducing expressions into their binomial factors, students can simplify complex problems, paving the way for more advanced mathematical concepts.