Problem 35 - A Graphical Approach to College Algebra

Question

Factor each perfect square trinomial completely. $$20 p^{2}-100 p q+125 q^{2}$$

Step-by-Step Solution

Answer

The factored form is \((2\sqrt{5}p - 5\sqrt{5}q)^2\)."

Step 1: Identify the perfect square trinomial

Recognize that the given trinomial is in the form of a perfect square trinomial. A perfect square trinomial can generally be written as: \[ a^2 - 2ab + b^2 = (a - b)^2 \] where \( a \) and \( b \) are expressions. Our task is to express \( 20p^2 - 100pq + 125q^2 \) in this form if possible.

Step 2: Take the square roots of the first and last terms

Identify \( a^2 = 20p^2 \) and \( b^2 = 125q^2 \). Calculate: - \( a = \sqrt{20p^2} = \sqrt{4\cdot5p^2} = 2\sqrt{5}p \) - \( b = \sqrt{125q^2} = \sqrt{25\cdot5q^2} = 5\sqrt{5}q \) These calculations determine potential values for \( a \) and \( b \).

Step 3: Verify the middle term

For the trinomial to be a perfect square, the middle term should be \(-2ab\). Calculate \( -2ab \):\[-2(2\sqrt{5}p)(5\sqrt{5}q) = -2\times10pq\times5 = -100pq \]This confirms that the middle term \(-100pq\) matches \(-2ab\), verifying the trinomial is a perfect square.

Step 4: Write the trinomial as a square of a binomial

Since all conditions are satisfied, express the trinomial as the square of a binomial:\[ (2\sqrt{5}p - 5\sqrt{5}q)^2 \] This is the factored form of the perfect square trinomial \( 20p^2 - 100pq + 125q^2 \).

Key Concepts

Factoring Trinomials

Factoring trinomials is an essential algebraic technique used to simplify expressions or solve equations. A trinomial is a polynomial with three terms. The process of factoring involves rewriting the trinomial as a product of simpler expressions. This often entails breaking it down into binomials. To factor a trinomial like a perfect square trinomial, one must first identify its specific form.

A perfect square trinomial looks like this:

• \( a^2 - 2ab + b^2 \)

This can be rewritten as \((a - b)^2\). Factoring perfect square trinomials involves confirming that the middle term equals \(-2ab\), serving as a check for whether the expression can be transformed into the square of a binomial. By following these steps, you can seamlessly factor many algebraic expressions, simplifying them for further calculations or interpretations.

Binomial Squaring

Binomial squaring is a process that involves multiplying a binomial by itself. This often results in the formation of perfect square trinomials, which are quite recognizable due to their structure. When a binomial \((a - b)\) is squared, it unfolds into a perfect square trinomial through the distributive property. Let's break it down:

• The binomial \((a - b)^2\) is expanded to \( (a - b)(a - b) \).
• Use the distributive property to multiply: \( a(a - b) - b(a - b) \).
• This results in \( a^2 - 2ab + b^2 \).

Understanding how a binomial squares into a trinomial helps in recognizing and factoring expressions. It is crucial to always verify the middle term \(-2ab\) to ensure it matches correctly when trying to express a trinomial as a binomial squared.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and coefficients combined through arithmetic operations. At their core, they are the building blocks of algebra. The expression \( 20p^2 - 100pq + 125q^2 \) is a type of algebraic expression known as a trinomial because it has three terms. Breaking down algebraic expressions into simpler components, such as binomials, makes them easier to work with and understand

• Variables: Symbols like \( p \) and \( q \) that represent numbers.
• Constants: Specific numbers like 20, 100, and 125 in the expression.
• Coefficients: Numbers multiplying the variables, such as 20 in \( 20p^2 \).

Algebraic expressions can be manipulated and simplified using techniques like factoring, which allows for solving equations and understanding the relationships between variables. Mastering these expressions equips you with tools to decipher more complex algebraic problems.