Problem 54 - Algebra 2

Question

CHECK FACTORING. Use a graphing calculator to determine if each polynomial is factored correctly. Write yes or no. If the polynomial is not factored correctly, find the correct factorization. $$ 3 x^{2}-48 \stackrel{?}{=} 3(x+4)(x-4) $$

Step-by-Step Solution

Answer

Yes, the polynomial is factored correctly.

Step 1: Expand the Given Factors

Start by expanding the factors on the right side to see if they match the original polynomial. Recall that to multiply two binomials, you use the distributive property. Here, multiply \((x+4)(x-4)\).

Step 2: Calculate the Binomial Product

Calculate \((x+4)(x-4)\) using the formula for the difference of squares: \[(a+b)(a-b) = a^2 - b^2\]Applying to our binomials: \((x+4)(x-4) = x^2 - 16\).

Step 3: Multiply by the Coefficient

Now multiply the product by 3:\[3(x^2 - 16) = 3x^2 - 48\].

Step 4: Compare With the Original Polynomial

Compare the expansion \(3x^2 - 48\) with the original polynomial \(3x^2 - 48\). Since they match, the factorization is correct.

Key Concepts

Difference of Squares

The Difference of Squares is a special pattern in algebra used to factor certain types of polynomials. It is expressed as \( a^2 - b^2 \), and it factors into \( (a+b)(a-b) \). This is extremely useful when you come across a polynomial that can be written in the form of two perfect squares separated by a subtraction sign. For example, in the expression \((x+4)(x-4)\), we recognize \(x^2\) and \(4^2\) as our perfect squares. Therefore, \( x^2 - 16 \) is an excellent example of the difference of squares. Applying this formula makes factoring quick and often reduces the complexity of solving algebraic problems.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler factors. This is similar to factoring numbers, where you express a number as a multiplication of its prime factors. In algebra, by factoring, we simplify complex polynomial expressions into simpler ones.

Some key purposes of factorization include:

• Solving polynomial equations more easily.
• Simplifying expressions for analysis.
• Helping in understanding the relationships within polynomials.

Always check if your factorization is correct by expanding it back to the original polynomial like it was done in the original solution.

Graphing Calculator Usage

Graphing calculators are powerful tools in verifying the factorization of polynomials. They allow you to graph the polynomial's expression and its factors, helping you visually confirm or identify errors. Here's how you can use it effectively:

• Input the original polynomial \(3x^2 - 48\).
• Input the factored form \(3(x+4)(x-4)\).
• Graph both expressions.
• Check if both graphs overlap completely, indicating accurate factorization.

Using a graphing calculator provides a visual way to verify your solution, making it easier to understand the behavior of the polynomial.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and algebraic operations. These expressions form the core of algebra, as they are used to represent relationships and patterns.

Key components include:

• Variables represented by letters like \(x\) or \(y\) that take on different values.
• Constants, which are fixed values such as numbers.
• Operations including addition, subtraction, multiplication, and division.

When working with expressions like \(3x^2 - 48\), you often need to perform operations such as factoring to simplify and solve problems. Understanding how to manipulate and factor these expressions is key to mastering algebra.