Question

Challenge Problems.$$4 m^{2}-9 n^{4}$$

Short answer

\(4m^2 - 9n^4 = (2m + 3n^2)(2m - 3n^2)\)

Step-by-step solution

Identify the Expression Type

Identify the given expression, which is a difference of squares. A difference of squares is an algebraic form that can be factored into the form of \(a^2 - b^2 = (a + b)(a - b)\).

Identify the Terms to Be Factored

In the expression \(4m^2 - 9n^4\), identify \(a^2 = 4m^2\) and \(b^2 = 9n^4\). Taking the square root of each term, we find that \(a = 2m\) and \(b = 3n^2\).

Apply the Difference of Squares Formula

Now, apply the difference of squares formula. Thus, \(4m^2 - 9n^4 = (2m)^2 - (3n^2)^2 = (2m + 3n^2)(2m - 3n^2)\).

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that allows us to simplify expressions and solve equations. It involves breaking down a polynomial into its component factors, much like finding the prime factors of a number. A polynomial is an algebraic expression consisting of variables raised to various powers and multiplied by coefficients.

Take our example \(4m^2 - 9n^4\). It's a binomial because it has two terms, and it can be factored by recognizing it as a difference of squares. The process begins by identifying patterns that fit specific factoring formulas. For instance, if you have an expression like \(a^2 - b^2\), this is a classic 'difference of squares' pattern that can be factored into \(a+b)(a-b)\), where \(a\) and \(b\) are the square roots of the original terms.

By mastering the technique of factoring polynomials through recognizing patterns and applying the appropriate formulas, algebra becomes a much more manageable subject.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. In our case, \(4m^2 - 9n^4\) is an algebraic expression consisting of two terms that are subtracted from one another. Each term represents a mathematical object that can include whole numbers, variables (like \(m\) and \(n\) here), and exponents.

Understanding how to manipulate these expressions is fundamental in algebra. This might involve simplifying the expressions by combining like terms, or by factoring as in our example. Recognizing different forms of algebraic expressions, such as the difference of squares, simplifies the factoring process and is vital for progressing in algebra.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In algebra, we're often interested in the square root of variables with exponents, as these allow us to simplify expressions like the difference of squares.

In our textbook problem, we looked for the square root of each term in the binomial \(4m^2 \) and \(9n^4\) to apply the difference of squares identity. The square root of the first term, \(4m^2\), is \(2m\) and the square root of the second term, \(9n^4\), is \(3n^2\). Knowing how to find square roots is essential for factoring polynomials and for solving quadratic equations, among other applications in algebra.

Algebraic Identity

Algebraic identities are equations that hold true for all values of the variables involved. They are the backbone of simplifying algebraic expressions and solving equations. One such identity is the difference of squares, which states that \(a^2 - b^2 = (a + b)(a - b)\).

This identity is a powerful tool when working with polynomials. As in our exercise, \(4m^2 - 9n^4\) conforms to the difference of squares identity, with \(a\) and \(b\) being the square root of each term. Once we recognize this, we can apply the identity directly to factor the expression. Knowledge of algebraic identities greatly simplifies working with polynomials, making it easier to factor them, to solve equations, and to simplify algebraic expressions.