Problem 6 - Precalculus Mathematics for Calculus

Question

The Special Factoring Formula for a "perfect square" is \(A^{2}+2 A B+B^{2}=______\quad .\) So \(x^{2}+10 x+25\) factors as______.

Step-by-Step Solution

Answer

The trinomial factors as \((x + 5)^{2}\).

Step 1: Identify the Formula

The formula provided, \(A^{2}+2AB+B^{2}\), is known as the perfect square trinomial formula. It can be rewritten based on its factors as \((A+B)^{2}\). This means that a trinomial in the form \(A^{2}+2AB+B^{2}\) can be factored into a binomial squared.

Step 2: Match the Trinomial with the Formula

The given trinomial is \(x^{2} + 10x + 25\). We compare this with \(A^{2} + 2AB + B^{2}\) to identify \(A\) and \(B\). Here, \(x^{2}\) matches \(A^{2}\), indicating \(A = x\), and \(25\) is \(B^{2}\), so \(B = 5\) because \(5^2 = 25\).

Step 3: Verify the Middle Term

To ensure correctness, verify the middle term \(2AB\). We found \(A = x\) and \(B = 5\). Calculating \(2AB\) gives \(2 \cdot x \cdot 5 = 10x\), which matches the middle term of the trinomial \(x^{2} + 10x + 25\).

Step 4: Write the Factored Trinomial

Since all conditions match, the trinomial \(x^{2} + 10x + 25\) factors according to the perfect square trinomial formula as \((x + 5)^{2}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of algebraic expression. It takes the form of \(A^2 + 2AB + B^2\). The special thing about it is that such trinomials can be factored neatly into a binomial squared. To break it down:

• \(A^2\) is the square of the first term.
• \(B^2\) is the square of the second term.
• \(2AB\) accounts for two times the product of \(A\) and \(B\).

The beauty of perfect square trinomials is their symmetry. They simplify down to \((A + B)^2\). Understanding this pattern is useful for quickly identifying potential perfect squares in algebraic problems. Think of it like recognizing a friend in a crowd by spotting their distinctive features. This makes your life much easier when dealing with algebra!
Identifying these trinomials helps in simplifying expressions or solving equations effectively.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). They form the language of algebra, allowing us to express mathematical ideas concisely.

• Each part of an expression (like \(x^2\), \(10x\), and \(25\) in \(x^2 + 10x + 25\)) is called a term.
• Expressions can be simplified by combining like terms or by factoring.

When working with algebraic expressions, it's like building a puzzle. You move pieces around until they fit into a pattern that makes sense. This pattern might be a perfect square trinomial, a factorable quadratic, or another recognizable form.
Manipulating these expressions allows us to solve equations, find patterns, and reveal insights about broader mathematical systems. Whether recognizing a form or factoring a trinomial, knowing how to handle algebraic expressions is key to mastering algebra.

Binomial Squares

A binomial square is the result of squaring a binomial algebraic expression. When you square a binomial, like \((A + B)^2\), you expand it into a trinomial. The expanded form is exactly our perfect square trinomial: \(A^2 + 2AB + B^2\).

• Start with a binomial, say \((x + 5)\).
• Squaring it gives \((x + 5)^2\).
• Upon expansion, you get the terms: \(x^2 + 10x + 25\).

This link between binomials and trinomials is crucial. It's like seeing the same object from different angles. By understanding that a binomial square always results in a perfect square trinomial, you can reverse the process too. So, if you see \(x^2 + 10x + 25\), you know it factors back into \((x + 5)^2\).
Recognizing and working with binomial squares helps in simplifying expressions and solving equations, making algebra more approachable and fun!