Question

Factor the polynomial completely. $$ 4 m^4-25 $$

Short answer

The completely factorized form of \(4m^4 - 25\) is \((2m^2 - 5)(2m^2 + 5)\).

Step-by-step solution

Factor The Binomial As A Difference Of Squares

The polynomial \(4m^4 - 25\) is the difference of two squares. The general form of difference of squares is \(a^2 - b^2\) which factors into \((a - b)(a + b)\). So, first, we need to write \(4m^4 - 25\) in the form of \(a^2 - b^2\). And it can be written as \((2m^2)^2 - (5)^2\).

Factorize the Diffence of Squares

Now we apply the difference of squares factorization. So, we factorize \((2m^2)^2 - (5)^2\) as two terms \((2m^2 - 5)(2m^2 + 5)\).

Final Result

So, the given binomial is factored completely as \((2m^2 - 5)(2m^2 + 5)\)

Key concepts

Difference of Squares

The difference of squares is a special algebraic technique used in factoring. It applies to expressions that can be written in the form \(a^2 - b^2\). Here, you will transform these into a product of two binomials: \((a - b)(a + b)\).
To use this method, the given expression must really be a difference (indicating subtraction) and consist of two perfect squares. In the example given, the expression \(4m^4 - 25\) can be converted into \((2m^2)^2 - 5^2\), confirming it as a difference of squares. This allows you to apply the simple transformation to convert it into \((2m^2 - 5)(2m^2 + 5)\).

Recognizing these patterns takes practice, but once you can identify an expression as a difference of squares, it makes factoring significantly easier.

Polynomials

Polynomials are mathematical expressions involving sums of powers of variables with coefficients. They can range from simple expressions, like \(x + 2\), to more complex terms, such as the \(4m^4\) in the expression \(4m^4 - 25\).

A polynomial is usually represented in the form:

• The coefficient \(a_n\) are constants, and must not all be zero.
• The powers of the variable are the exponents and must be non-negative integers.
• The specific number of terms determines the degree of the polynomial.

In our given problem, \(4m^4 - 25\) is simplified and distinguished because it consists of a quadratic term and a constant, allowing us to employ the difference of squares technique for factoring.

Algebraic Expressions

Algebraic expressions describe equations or expressions composed of numbers, variables, and operations. They can be as simple as a single number or variable or more complex like \(4m^4 - 25\).

The key features of algebraic expressions include:

• They can include constants (e.g., numbers like 25 in the example).
• They often contain variables (like \(m\) in the problem above) that stand in for unknown or any real number.
• These expressions can be manipulated, combined, and factored depending on their form or situation.

Understanding algebraic expressions is essential when learning to factor, as it involves identifying patterns within these structures to simplify or rearrange them. As seen in the exercise, recognizing that \(4m^4 - 25\) is an algebraic expression allowed us to use known formulas, such as the difference of squares, to simplify and factor it completely.