Problem 30 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 10 y^{2}-320 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given binomial 10y^2 - 320 is 10(y + 4√2)(y - 4√2).

Step 1: Identify the Greatest Common Factor (GCF)

Find the greatest common factor of the terms 10y^2 and -320. The GCF of 10y^2 and 320 is 10 as both terms are divisible by 10.

Step 2: Factor out the GCF

Factor the GCF (10) out of the binomial. 10(y^2 - 32)

Step 3: Factor the remaining expression (Difference of squares)

Now, we're left with the expression y^2 - 32. We can factor this as a difference of squares (a^2 - b^2 = (a+b)(a-b)) with a = y and b = 4√2 (since 32 = (4√2)^2): y^2 - 32 = (y + 4√2)(y - 4√2)

Step 4: Write the final factored form

Combine the GCF obtained in step 2 with the factored expression from step 3: 10y^2 - 320 = 10(y + 4√2)(y - 4√2)

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is essential to simplifying algebraic expressions. The GCF is the largest number that divides evenly into each term of an expression. For example, when factoring the binomial 10y^2 - 320, we look for the highest number that both 10y^2 and 320 have in common. In this case, 10 is the GCF since both terms can be divided by 10 without leaving a remainder.

Factoring out the GCF simplifies the expression and makes further factoring steps more manageable. In practice, you would divide each term by 10, which gives us a new expression inside the brackets, y^2 - 32. This step is pivotal as it paves the way for employing other factoring methods such as the difference of squares, which we will explore in the following section.

Difference of Squares

The difference of squares is a pattern in algebra that is used to factor certain types of binomials. Specifically, it applies to binomials where each term is a square and they are subtracted from one another, thus the name 'difference of squares'. The formula is \[ a^2 - b^2 = (a + b)(a - b) \.\]

When we encounter \( y^2 - 32 \) from our previous result, we identify it as a difference of squares. Here, \( y^2 \) is our \( a^2 \) and \( 32 \) can be rewritten as \( (4\sqrt{2})^2 \) which serves as our \( b^2 \) in the formula. Thus, the expression can be factored into \( (y + 4\sqrt{2})(y - 4\sqrt{2}) \).

Recognizing this pattern helps in breaking down complex expressions into simpler, multiplicative factors, which is a common objective in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (such as \( y \) in our example), and arithmetic operators like addition and subtraction. Factoring is a method to write these expressions as a product of simpler expressions. The beauty of algebra is in these simplifications, which can make seemingly difficult problems much more approachable.

With the expression \( 10y^2 - 320 \), we applied our knowledge of the greatest common factor and the difference of squares. The final factored form of this binomial is \( 10(y + 4\sqrt{2})(y - 4\sqrt{2}) \) which demonstrates how we've deconstructed a complex expression into a product of simpler ones. This is not just a mechanical process; it's an insight into the structure of algebraic expressions that can aid in solving equations, graphing functions, and understanding the relationships between algebraic quantities.