Problem 57 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}-10 x+100$$

Step-by-Step Solution

Answer

The factored form of the given perfect square trinomial \(x^{2}-10x+100\) is \((x-10)^{2}\).

Step 1: Identifying the pattern

First, the trinomial is compared with the formula \(a^{2}±2ab+b^{2}\). After reviewing the trinomial \(x^{2}-10x+100\), it can be noticed that it fits the formula where \(a=x\), \(b=10\), and \(-2ab=-10x\) which is the middle term, and \(b^{2}=100\) which is the third term.

Step 2: Factoring the trinomial

Since it fits the pattern of a perfect square trinomial, it can be factored into \((a-b)^{2}\), replacing \(a\) and \(b\) with \(x\) and \(10\), respectively. We then get the factored form, which is \((x-10)^{2}\).

Step 3: Final answer

Therefore, the factored form of the given perfect square trinomial is \((x-10)^{2}\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a specific type of polynomial. It takes the form \(a^2 \pm 2ab + b^2\). If a trinomial matches this pattern, you can rewrite it as a squared binomial. Identifying a perfect square trinomial involves looking at three terms and recognizing them as squares and a double product. It is like rearranging a puzzle, where you see connections.

• The first term should be a perfect square: like \(x^2\) in our exercise.
• The third term should also be a perfect square: like \(100\), where \(10^2 = 100\).
• The middle term should equal twice the product of numbers corresponding to the first and third terms' square roots. In this case, \(-10x\) is twice the product of \(x\) and \(10\).

By confirming these conditions, we've identified our trinomial as a perfect square. It can then be easily factored into the form \((x-b)^2\), making it very approachable to factor.

Factoring Techniques

Factoring is a technique used to express a polynomial as the product of its roots or terms. There are various methods for factoring, but identifying special patterns often simplifies the process. One of these patterns is the perfect square trinomial, already discussed.
In general:

• Look for common factors. If each term in a polynomial shares a factor, extract it first.
• Check if it matches a recognizable pattern, like the perfect square trinomial or difference of squares.
• Decompose middle terms to make factoring by grouping possible.

Using these techniques helps break down complex polynomials into simpler, more manageable parts. This makes solving equations easier. By recognizing our original problem as a perfect square trinomial, we used the pattern recognition technique for efficient factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations like addition and multiplication. They resemble phrases in regular language but speak the detailed language of mathematics. Understanding algebraic expressions is fundamental to learning algebra.

In the context of factoring:

• Recognizing terms and coefficients is crucial. Like in our trinomial \(x^2 - 10x + 100\), each part has a role.
• The process of factoring reorganizes an expression. It identifies simpler expressions (called factors) whose product returns the original expression.
• It forms a foundation for solving equations and simplifying mathematical problems.

Whether simple or complex, mastering the understanding of algebraic expressions helps unlock more advanced ideas in mathematics. By practicing problems involving perfect square trinomials and other techniques, students strengthen their algebraic skillset.