Problem 32 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(y-4)^{2}$$

Step-by-Step Solution

Answer

The expanded form is $y^2 - 8y + 16$..

Step 1: Recognize the Pattern

The exercise $(y-4)^2$ represents a special case of multiplying binomials known as a perfect square trinomial. The formula for squaring a binomial $(a - b)^2$ is given by $a^2 - 2ab + b^2$. In this problem, $a = y$ and $b = 4$.

Step 2: Apply the Formula

Now, apply the formula $(a - b)^2 = a^2 - 2ab + b^2$ to the binomial $(y - 4)^2$. Substitute $a = y$ and $b = 4$ into the formula: $(y - 4)^2 = y^2 - 2(y)(4) + 4^2$.

Step 3: Calculate Each Term

Calculate each term in the expression: - $a^2 = y^2$. - $2ab = 2(y)(4) = 8y$. - $b^2 = 4^2 = 16$.

Step 4: Write the Expanded Form

Substitute the calculated values into the expression $y^2 - 8y + 16$ to expand the binomial product. This is the expanded form of $(y - 4)^2$.

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are a special form of trinomials derived from squaring a binomial, following a specific pattern. When you encounter an expression like \((a - b)^2\), it automatically falls into the category of perfect square trinomials. This means you can directly expand it using the formula: \((a - b)^2 = a^2 - 2ab + b^2\).

• First term, \(a^2\): Square the first term of the binomial.
• Middle term, \(-2ab\): Multiply the first term by the second term and then multiply by 2.
• Last term, \(b^2\): Square the second term of the binomial.

This formula is not only a significant shortcut, but helps avoid errors when expanding such expressions. Recognizing a perfect square trinomial saves time and simplifies the process.

Multiplying Binomials

Multiplying binomials involves using the distributive property twice. This method, often referred to as FOIL, stands for First, Outer, Inner, Last. It helps you remember which terms to multiply together. However, recognizing special patterns like perfect square trinomials can streamline the process.To illustrate the FOIL method:- **First:** Multiply the first terms from both binomials.- **Outer:** Multiply the outer terms (the first term of the first binomial and the last term of the second).- **Inner:** Multiply the inner terms (the last term of the first binomial and the first term of the second).- **Last:** Multiply the last terms of both binomials.After multiplying, combine like terms to simplify the expression. In expressions like \(y - 4)^2\), instead of using FOIL, recognizing it as a perfect square trinomial simplifies it to a \(a^2 - 2ab + b^2\) pattern.

Binomial Expansion

Binomial expansion refers to the process of expanding an expression that involves a power of a binomial, like \(a + b)^n\). For powers higher than 2, you often use the Binomial Theorem.However, when \(n = 2\), the expansion can follow recognizable patterns like perfect square trinomials. This makes binomial expansion simpler because:- You only need to use basic algebraic identities, such as \((a - b)^2 = a^2 - 2ab + b^2\).- It avoids the need to apply the entire Binomial Theorem.Recognizing these patterns helps simplify expressions without excessive calculations. This is especially useful in calculus and algebra, where binomial expansions frequently occur.