Problem 59 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$20 a^{4}-45 a^{2}$$

Step-by-Step Solution

Answer

The factorized form of the given polynomial \(20a^{4} - 45a^{2}\) is \(5a^2(2a+3)(2a-3)\).

Step 1: Identify and Factor out the Greatest Common Factor

First, identify the greatest common factor (GCF) between the terms in the polynomial. Here the GCF is \(5a^2\). So, pull out this from each term of the polynomial to get: \(5a^2(4a^2 - 9)\)

Step 2: Apply Difference of Squares

The new polynomial \(4a^2 - 9\) is a difference of squares, i.e., it can be written in the form of \(a^2-b^2\) which is equal to \((a+b)(a-b)\). Here, apply this formula where \(a = 2a\) and \(b = 3\). So, the expression becomes: \(5a^2((2a)^2 - (3)^2)\) equals \(5a^2((2a+3)(2a-3))\).

Step 3: Write the final factorized form

Putting all together, express the factorized form of the polynomial: The final factorized form is \(5a^2(2a+3)(2a-3)\).

Key Concepts

Greatest Common Factor

Factoring polynomials often starts with identifying the Greatest Common Factor (GCF). The GCF is the largest expression that divides each term of the polynomial without a remainder.
In the given polynomial, \(20a^4 - 45a^2\), we first look at the numerical coefficients and the common variable part.

• For the coefficients 20 and 45, the greatest common factor is 5.
• For the variables \(a^4\) and \(a^2\), since they both have the term \(a^2\), that becomes the greatest power of \(a\) common to both terms.

Thus, the GCF of the entire polynomial is \(5a^2\). Once identified, factor out this GCF from each term, simplifying the expression as follows:\[20a^4 - 45a^2 = 5a^2(4a^2 - 9)\]By factoring out the GCF, you simplify the polynomial, making it easier to apply other techniques like the difference of squares.

Difference of Squares

The next step in factoring is to recognize if the remaining polynomial can be expressed as a difference of squares. This method is particular to expressions of the form \(a^2 - b^2\), which can be factorized as \((a+b)(a-b)\).
In the partially factorized expression \(4a^2 - 9\), we notice:

• \(4a^2 = (2a)^2\)
• \(9 = 3^2\)

Therefore, \(4a^2 - 9\) can be written as:\[(2a)^2 - 3^2\]Using the difference of squares formula, we factor it as:\[(2a + 3)(2a - 3)\]By applying the difference of squares, the expression becomes simpler, allowing us to see its two binomial components clearly. This is an invaluable technique in algebra for making complex expressions more manageable.

Algebraic Expressions

Understanding algebraic expressions is key in the journey of factoring polynomials. An algebraic expression is a mathematical phrase that can involve numbers, variables, and operations.
The original exercise started with the expression $20a^4 - 45a^2$. This is an example of a polynomial, which is a type of algebraic expression consisting of terms combined with addition or subtraction.
When working with polynomials:

• Each term is a product of a coefficient and a variable raised to a power.
• The degree of each term is determined by the power of the variable.
• Simplifying and factoring help us rewrite expressions in simpler, more useful forms.

By recognizing factors and correct formulas, algebraic expressions can be transformed, aiding in solving equations and simplifying math problems. This process is a fundamental aspect of algebra that often leads to deeper insights into the expressions and their behaviors.