Problem 39 - College Algebra

Question

In Exercises \(31-40,\) factor the difference of two squares. $$16 x^{4}-81$$

Step-by-Step Solution

Answer

The factored form of \(16 x^{4}-81\) is \((4x^2 + 9)(2x + 3) (2x - 3)\)

Step 1: Identify terms

First it should be recognized that the equation \(16 x^{4}-81\) is a difference of squares. Notice how each term is a perfect square: \(16x^4\) is \((4x^2)^2\) and \(81\) is \(9^2\)

Step 2: Apply the formula

Apply the formula for factoring difference of square \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a = 4x^2\) and \(b = 9\). So, the factored form will be \((4x^2 + 9)(4x^2 - 9)\)

Step 3: Factorise further if possible

Upon examination of the factors, \((4x^2 - 9)\) it can still be factored further. We can apply the formula again to get \( (2x + 3) (2x - 3)\) as the factors of \((4x^2 - 9)\)

Step 4: Final factored form

The complete factored form of the original equation \(16 x^{4}-81\) will be \((4x^2 + 9)(2x + 3) (2x - 3)\)

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is a key skill in algebra. Polynomials are expressions made up of terms, each consisting of a variable raised to an exponent, multiplied by a coefficient. Factoring is the process of breaking down a polynomial into simpler terms (or 'factors') that, when multiplied together, give the original polynomial.

To factor a polynomial, you often look for common factors among the terms or identify specific patterns, such as a difference of squares or perfect square trinomials. Once these patterns are recognized, you can apply suitable factoring formulas to simplify the polynomial. These steps make solving higher-level algebra problems more manageable.

Perfect Squares

Perfect squares are numbers or algebraic expressions that can be expressed as the square of an integer or a polynomial. For instance, 81 is a perfect square because it equals 9 squared ( 9^2 = 81 ). Similarly, in algebra, 16x^4 is a perfect square because it equals ( 4x^2 ) squared. Recognizing perfect squares is crucial because they often allow us to simplify complex algebraic expressions. In the difference of squares formula, perfect squares are the backbone. Factoring expressions like 16x^4 - 81 involves identifying these perfect squares as coefficients or terms and transforming them into a product of binomials.

Factoring Techniques

There are various factoring techniques used in algebra to simplify polynomials:

• Greatest Common Factor (GCF): This involves finding the largest factor shared by all terms in the polynomial and factoring it out.
• Difference of Squares: This technique is used when a polynomial can be expressed as a difference between two perfect squares. For example, using the formula (a^2 - b^2 = (a + b)(a - b) ) simplifies the expression.
• Trinomials: Factoring these involves finding two binomials that multiply to form the original trinomial, using techniques such as "trial and error" or "grouping."

Understanding these techniques enables you to tackle a variety of algebraic expressions effectively, paving the way for solving increasingly complex equations and functions.