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Chapter 12: Problem 63

Review factoring polynomials (Sections \(6.1-6.6)\) Factor $$ 3 x^{2}-48 $$

Short Answer

Expert verified
The factored form is \( 3(x + 4)(x - 4) \).

Step by step solution

01

- Identify the Common Factor

Observe the given polynomial. Notice that both terms, 3x^2 and -48, share a common factor, which is 3.
02

- Factor Out the Common Factor

Factor out the common factor, 3, from each term in the polynomial: \[ 3(x^2 - 16) \]
03

- Identify a Difference of Squares

The expression inside the parentheses, \( x^2 - 16 \), is a difference of squares. A difference of squares has the form \( a^2 - b^2 = (a + b)(a - b) \). In this case, \( x^2 - 16 \) can be written as \( x^2 - 4^2 \).
04

- Apply the Difference of Squares Formula

Apply the difference of squares formula to factor \( x^2 - 4^2 \): \[ x^2 - 4^2 = (x + 4)(x - 4) \]
05

- Write the Final Factored Form

Combine the factored terms from Steps 2 and 4 to write the fully factored form of the original polynomial: \[ 3(x + 4)(x - 4) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common factor
To begin factoring polynomials, it's crucial to look for a common factor in all terms of the polynomial. A common factor is a number or variable that evenly divides each term.
In the given problem, we have the polynomial **3x^2 - 48**.
  • Identify the common factor by examining the coefficients (numbers in front of the variables) and any variables (like x) present in all terms.
  • In this case, both terms (3x^2 and -48) share the common factor of 3.
  • So, we factor out the 3:
    $$3(x^2 - 16)$$
Factoring out the common factor simplifies the expression and makes it easier to tackle remaining factors.
difference of squares
Next, let's consider the expression inside the parentheses: **x^2 - 16**.
This represents a special type of polynomial called a difference of squares.
The general form for a difference of squares is **a^2 - b^2**, which can be factored into:
$$a^2 - b^2 = (a + b)(a - b)$$
In our case, we have x^2 - 16, which can be rewritten as x^2 - 4^2. Here, a = x and b = 4.
Now applying the difference of squares formula, we get:
$$x^2 - 4^2 = (x + 4)(x - 4)$$
Remember, recognizing patterns such as the difference of squares can help simplify factoring and solve polynomial equations more efficiently.
factored form
With all parts factored, our last step is to combine these factors.
From our previous work, we have:
$$ 3(x^2 - 16) $$ transformed to $$ 3(x + 4)(x - 4) $$
This is the factored form of the original polynomial.
Each component plays a role:
  • The number 3 is the common factor we factored out initially.
  • The terms (x + 4) and (x - 4) come from using the difference of squares method to factor x^2 - 4^2.

Understanding how to convert the polynomial into its completely factored form helps in simplifying equations and solving algebraic expressions.

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