Problem 58 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$5 x^{3}-45 x$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial equation is: \(5x * (x + 3)(x - 3)\).

Step 1: Identify common factors

Examine the given polynomial, \(5x^{3} - 45x\). The key is to notice that both terms of the equation share the common factors x and 5.

Step 2: Factor out the common factor

Take out the common factors from both terms in the equation which results in the factorized version of the polynomial. This results in: \(5x * (x^{2} - 9)\)

Step 3: Further simplify

The expression within the brackets can be further factorized, as it resembles the standard form of difference of squares which factors into \((a + b)(a - b)\). Here, \(x^{2} - 9\) can be further simplified to \((x + 3)(x - 3)\).

Key Concepts

Common Factor

When working with polynomials, identifying a common factor is the first step towards simplifying any given expression. A common factor is a number or variable that divides all terms of a polynomial without leaving a remainder. In the polynomial you are given, which is \(5x^3 - 45x\), the common factor is \(5x\).

These terms, \(5x^3\) and \(-45x\), both contain \(5x\) as a factor, making it possible to extract this common factor and rewrite the polynomial in a simpler form. Once you remove \(5x\), the expression inside the parenthesis becomes \(x^2 - 9\).

Remember, not all polynomials have a common factor, but when they do, pulling it out is a crucial first step. It's like opening a door to other potential simplifications!

Difference of Squares

Difference of Squares is a specific pattern within polynomials that allows further simplification. This pattern follows the formula \(a^2 - b^2 = (a + b)(a - b)\). It represents factors that when multiplied back together provide the original terms separated by subtraction.

In the expression \(x^2 - 9\), you can identify a difference of squares pattern, because \(x^2\) is a perfect square (\(x^2 = x \times x\)) and \(9\) is also a perfect square (\(9 = 3 \times 3\)). The expression matches the pattern, with \(a = x\) and \(b = 3\).

• \((x^2 - 9)\) becomes \((x + 3)(x - 3)\).

Factoring a difference of squares not only simplifies the polynomial but also makes it easier to understand and further manipulate algebraically. Keep an eye out for this pattern; it's frequent in polynomial expressions and very useful for simplification.

Polynomial Simplification

Polynomial simplification is a process of reducing a polynomial to its simplest form. This involves factoring out common terms and recognizing patterns such as the difference of squares to further break down the expression.

The provided polynomial, \(5x^3 - 45x\), simplifies further through examination of its factors. Extracting the common factor \(5x\), as explained earlier, transforms it to \(5x(x^2 - 9)\). Then, identifying \(x^2 - 9\) as a difference of squares allows it to be factored to \((x + 3)(x - 3)\). This makes the fully simplified version of the polynomial: \(5x(x + 3)(x - 3)\).

• Simplifying polynomials helps in solving equations, finding roots, and analyzing functions.
• Always check for common factors first, then look for recognizable patterns.

Simplification is vital in mathematics because it reduces complexity making further calculations less error-prone and more intuitive.