Problem 31 - 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. $$(y-7)^{2}$$

Step-by-Step Solution

Answer

The expanded form is \( y^2 - 14y + 49 \).

Step 1: Identify the Pattern

Recognize that the expression \( (y - 7)^2 \) is a perfect square trinomial. The general formula for the square of a binomial is \( (a - b)^2 = a^2 - 2ab + b^2 \).

Step 2: Apply the Formula

In the expression \( (y - 7)^2 \), identify \( a = y \) and \( b = 7 \). Apply the formula for the square of the binomial: \( a^2 - 2ab + b^2 \).

Step 3: Calculate Each Term

Calculate \( a^2 = y^2 \), \(-2ab = -2 imes y imes 7 = -14y\), and \( b^2 = 7^2 = 49 \).

Step 4: Write the Expanded Expression

Combine the calculated terms to write the expanded expression: \( y^2 - 14y + 49 \).

Key Concepts

Perfect Square Trinomial

When multiplying binomials, especially when squaring a binomial, you often encounter a special pattern known as the perfect square trinomial. This arises when you have an expression of the form \((a - b)^2\) or \((a + b)^2\). Recognizing this pattern is key:

• For \((a - b)^2\), the expression expands into a perfect square trinomial: \(a^2 - 2ab + b^2\).
• Similarly, for \((a + b)^2\), it becomes \(a^2 + 2ab + b^2\).

This pattern reveals that a perfect square trinomial results in three specific terms: the square of the first binomial term \(a\), twice the product of the two terms \(a\) and \(b\), and the square of the second term \(b\). Understanding this concept can simplify problem solving, as it allows you to quickly expand and solve without processing through each individual multiplication step every time.
Instead, just identify your \(a\) and \(b\), plug them into the formula, and automatically arrive at your trinomial.

Binomial Expansion

The concept of binomial expansion is key when working with expressions like \((y - 7)^2\). It involves systematically expanding the product of two binomials, whether they are identical, as with a square, or slightly different.

• When you have an expression \((x + y)^n\), binomial expansion uses the binomial theorem to distribute each term, creating a sum of terms.
• The theorem itself states that you can expand \((a + b)^n\) into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
• For squaring a binomial like \((y - 7)^2\), the expansion can be efficiently achieved using the aforementioned perfect square trinomial result: \(y^2 - 14y + 49\).

This method allows for easily simplifying and understanding expressions without manually multiplying everything out. Binomial expansion is, thus, a fundamental part of algebra that aids in transforming algebraic expressions for further manipulation and analysis.

Algebraic Expressions

Algebraic expressions, like \((y - 7)^2\), are combinations of terms formed using numbers, variables, and arithmetic operations. They are foundational in mathematics and serve as representations of real-world scenarios or abstract concepts in a more flexible form.

• The components of algebraic expressions include variables (like \(y\)), constants (numerical values), and operation symbols (like "+" and "-").
• Operations on algebraic expressions often involve expanding, factoring, simplifying, or solving these expressions.
• When working with expressions like \((y - 7)^2\), understanding the operations involved, such as the exponent part of the expression, is crucial for speedy computation and accuracy.

Comprehending how these expressions can be transformed—such as expanding \((y - 7)^2\) into \(y^2 - 14y + 49\)—provides a platform for solving equations, graphing functions, and more complex mathematical tasks. Mastery of algebraic expressions opens the door to success in various mathematical fields and practical problem-solving.