Problem 24 - Elementary Algebra

Question

Find the following products and simplify. $$ (y-8)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplest form of the expression $(y-8)^2$ is $y^2 - 16y + 64$.

Step 1: Write down the expression and expand the product.

First, let's write down the given expression, which is: $$(y-8)^2$$ Now, expand the product as the multiplication of \((y-8)\) by itself: $$(y-8)(y-8)$$

Step 2: Apply FOIL method to multiply the binomials.

To multiply these binomials, we will apply the FOIL method. FOIL stands for First, Outer, Inner, and Last, which are the four products we need to calculate in this method. $$(y-8)(y-8) = (y*y) + (-8*y) + (y*-8) + (-8*-8)$$

Step 3: Simplify the expression.

Now, let's simplify the expression by performing the multiplication and simplification of terms. $$(y-8)(y-8) = y^2 - 8y - 8y + 64$$ Combine the like terms: $$y^2 - 16y + 64$$

Step 4: Write the final result.

The result of the product and simplification is as follows: $$(y-8)^2 = y^2 - 16y + 64$$

Key Concepts

FOIL method

Mastering the FOIL method is essential for success in algebra. FOIL is an acronym that stands for First, Outer, Inner, and Last, referring to the order in which you multiply terms when dealing with two binomials. For example, if you have \( (y-8)(y-8) \), the FOIL method helps you systematically expand the expression:

• First: Multiply the first terms in each binomial (\(y * y = y^2\)).
• Outer: Multiply the outer terms (\(y * -8 = -8y\)).
• Inner: Multiply the inner terms (\( -8 * y = -8y\)).
• Last: Multiply the last terms in each binomial (\( -8 * -8 = 64\)).

After applying the FOIL method, you combine these results to form a single expression. In our example, the FOIL method allows us to write \( (y-8)^2 = y^2 - 8y - 8y + 64 \).

Algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. Like the expression \( (y-8)^2 \) we're examining, they can represent a wide range of mathematical concepts. Understanding how to manipulate these expressions is a foundational skill in algebra.

Expressions can include exponents, like the squared term in our example, which signifies \(y\) multiplied by itself. Also, they may contain multiple terms, as seen after expanding our example using the FOIL method, resulting in \(y^2 - 16y + 64\). Familiarity with different types of expressions, such as polynomials, will aid in solving more complex equations and problems in algebra and beyond.

Simplifying expressions

When you simplify an algebraic expression, your goal is to make it as straightforward as possible. Post-application of the FOIL method, simplification involves combining like terms and eliminating any redundancies. Like terms are terms that have the same variable raised to the same power—like \( -8y \) and \( -8y \) in the product \(y^2 - 8y - 8y + 64\).

To simplify our example, we combine the middle terms:

• Combine like terms: \( -8y - 8y \) becomes \( -16y \).

This results in the simplified form \(y^2 - 16y + 64\). It's essential to always combine like terms where possible to achieve the simplest form of an expression, making it easier to understand and, eventually, to solve equations.