Problem 50 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 60 y^{2}-240 $$

Step-by-Step Solution

Answer

The factored form of the expression \(60 y^{2} - 240 \) is \(60(y - 2)(y + 2)\).

Step 1: Factor out common factor from the expression

The first step is to see if there are any common factors to both 60 and 240. In this case, the common factor is 60. Factoring out 60 from both terms gives \(60(y^{2} - 4)\).

Step 2: Recognize the difference of squares

Notice the expression within the bracket is a difference of squares, as it can be written in the form of \(a^{2} - b^{2}\), where a is 'y' and b is '2'. The formula to factor the difference of squares is \((a^{2}-b^{2}) = (a-b)(a+b)\).

Step 3: Factor the difference of squares

Following the formula from Step 2, the expression within the bracket can be factored as \((y-2)(y+2)\). Thus, the fully factored expression of \(60 y^{2}-240\) is \(60(y-2)(y+2)\).

Key Concepts

Difference of Squares

When dealing with polynomial expressions, one technique used for simplifying them is recognizing the 'difference of squares'. The difference of squares is a specific type of expression that can be identified by its form. If you have an expression like \(a^2 - b^2\), it is a difference of squares because it's the subtraction (or difference) of two perfect squares.

Two perfect squares are numbers or expressions like 1, 4, 9, 16, or \(a^2, b^2\), where each is the square of a whole number or variable. The special thing about the difference of squares is the way it factors. It will always split into two binomials based on this formula:

• \((a^2 - b^2) = (a - b)(a + b)\)

Observing this in our expression \(y^2 - 4\), which can be altered into \((y - 2)(y + 2)\), makes factoring straightforward.

Always be on the lookout for these expressions as they constantly appear in algebraic equations and factoring them helps simplify and solve many problems effectively.

Common Factors

Before diving into more complicated methods like differences of squares, it's crucial to check if there's a simpler approach—finding common factors. A common factor means identifying a number or variable that evenly divides each term in the expression.

Consider the expression \(60y^2 - 240\). Here, both 60 and 240 have common factors. The largest number dividing both terms in this case is 60. When we pull this factor out of each term, we simplify the problem considerably. Factoring out 60 from \(60y^2 - 240\) gives us \(60(y^2 - 4)\). This step reduces the expression and allows for more advanced factoring techniques to be applied on the simplified terms.

Recognizing common factors serves as the foundation upon which more complex factoring is built, paving the way for easier solutions.

Polynomial Factoring

Polynomial factoring is a method that breaks down expressions into simpler multipliable parts or factors. This method is widely used in solving algebraic equations and simplifying them to reduce complexity. The ultimate goal is to express the original polynomial as a product of simpler polynomials.

Using the step-by-step solution from the original exercise, we start by identifying a common factor. For \(60y^2 - 240\), it's 60, allowing us to factor it into \(60(y^2 - 4)\). Next comes recognizing patterns such as the difference of squares inside the parenthesis, further simplifying this expression to \(60(y - 2)(y + 2)\) by using the difference of squares method.

Practicing polynomial factoring helps not only in solving equations but also in analyzing the behavior of functions. By factoring, complex expressions become manageable and ready for further mathematical operations or evaluations.