Question

Factor the expression. $$16 x^{2}-25$$

Short answer

The simplified form of the expression \(16 x^{2}-25\) is \((4x - 5)(4x + 5)\).

Step-by-step solution

Identify the squares

The given expression is \(16 x^{2}-25\). The term \(16x^{2}\) can be rewritten as \((4x)^{2}\) because \(4x * 4x = 16x^{2}\), and the term \(25\) can be rewritten as \(5^{2}\) because \(5 * 5 = 25\). So, the given expression becomes \((4x)^2 - 5^2\).

Apply the formula for difference of squares

The formula for the difference of squares is \(a^{2} - b^{2} = (a - b) (a + b)\). Now, apply this formula to the expression, with \(a = 4x\) and \(b = 5\). Therefore, \((4x)^2 - 5^2\) factorizes to \((4x - 5) (4x + 5)\).

Write the final answer

After applying the difference of squares formula, we have two factors: \((4x - 5)\) and \((4x + 5)\). So, the factored form of the expression \(16x^{2} - 25\) is \((4x - 5)(4x + 5)\).

Key concepts

Difference of Squares

The concept of the difference of squares is a cornerstone in algebra that applies to expressions where two perfect squares are subtracted from one another. A perfect square is a number or expression that can be written as a number multiplied by itself, such as \(9 = 3^2\)).

The formula for factoring the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\)). This formula tells us that if we have an expression where something squared is subtracted from another squared term, we can rewrite this expression as the product of two binomials. One binomial contains the positive and negative versions of the original terms (without the squares), and the other binomial contains the same terms, but with the opposite signs.

For instance, with \(16x^2 - 25\)), we can see that \(16x^2\)) is the square of \(4x\)) and \(25\)) is the square of \(5\)). Using the difference of squares formula, we can then factor this expression into \(\(4x - 5\)\(4x + 5\))\)).

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial expression into the product of simpler expressions, such as binomials or other polynomials. This is often performed by identifying patterns or using formulas like the difference of squares, or methods such as grouping, the square of a binomial, or the sum or difference of cubes.

When factoring, it's crucial to look for the greatest common factor (GCF) first, then apply specific factoring patterns or methods suitable for the given polynomial. However, not all polynomials are factorable with rational coefficients. When they are, factoring is a useful tool for simplifying expressions and solving polynomial equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \(x\)), and operations (such as addition, subtraction, multiplication, and division). They don't have an equality sign, unlike equations. Understanding how to work with and simplify algebraic expressions, like by factoring, is an essential skill in algebra.

Manipulating algebraic expressions involves performing operations in accordance with the laws of algebra, such as the distributive property, to simplify or rearrange them. This simplification is especially important when solving equations or inequalities, as it can make the process more straightforward.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically written in the standard form \(ax^2 + bx + c = 0\)), where \('a\)), \('b\)), and \('c\)) are constants and \(x\)) is the variable to be solved for. Factoring is one method to solve these equations when they can be rewritten as the product of two binomials set equal to zero.

After factoring a quadratic equation, you apply the Zero Product Property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero. Thus, solving the factored equation involves setting each binomial factor equal to zero and solving for \(x\)), which can give you two potential solutions for the variable.