Problem 26 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 9 x^{2}-100 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given binomial expression is $$(3x + 10)(3x - 10)$$.

Step 1: Identify a^2 and b^2 in the expression

First, we need to identify the values of $$a^2$$ and $$b^2$$ in the given binomial. From the given expression, $$9x^2 - 100$$, we can identify $$a^2 = 9x^2$$ and $$b^2 = 100$$.

Step 2: Calculate the square root of a^2 and b^2

Next, we need to find the square roots of $$a^2$$ and $$b^2$$ in order to get the values of $$a$$ and $$b$$. The square root of $$9x^2$$ is $$3x$$, and the square root of $$100$$ is $$10$$. So, $$a = 3x$$ and $$b = 10$$.

Step 3: Apply the difference of squares formula

Now that we have the values of $$a$$ and $$b$$, we can apply the difference of squares formula to factor the given binomial. The formula is: $$a^2 - b^2 = (a + b)(a - b)$$ Plugging in the values we found earlier, we get: $$(3x)^2 - (10)^2 = (3x + 10)(3x - 10)$$

Step 4: Write the final factored form

The factored form of the given binomial, $$9x^2 - 100$$, is $$(3x + 10)(3x - 10)$$.

Key Concepts

Difference of Squares

The concept of the difference of squares is fundamental in algebra. It refers to an expression of the form \( a^2 - b^2 \), which can be factored into \( (a + b)(a - b) \). This powerful algebraic identity allows us to simplify expressions that seem complex at first glance.

To apply this method, we need to identify two perfect squares that are being subtracted. In the exercise \( 9x^{2}-100 \), \( 9x^2 \) is the square of \(3x\) and \(100\) is the square of \(10\). Recognizing these squares is the critical first step. After finding the square roots \(a = 3x\) and \(b = 10\), we use the formula to rewrite the original expression as a product of binomials, resulting in \( (3x + 10)(3x - 10) \).

This formula is particularly useful because it appears frequently in various mathematical contexts, from factoring polynomials to solving equations.

Perfect Square Trinomials

A perfect square trinomial is an expression that can be factored into the square of a binomial. It takes the form of \( a^2 \pm 2ab + b^2 \), which factors to \( (a \pm b)^2 \). It's derived from squaring the binomial \( a \pm b \), which gives us the trinomial.

To recognize perfect square trinomials, we look for three terms: the first and last should be perfect squares, and the middle term should be twice the product of the square roots of the first and last terms. An example of this would be \( x^2 + 6x + 9 \), which factors to \( (x+3)^2 \).

While our exercise involves a binomial, not a trinomial, understanding perfect square trinomials is essential for more complex factoring and expanding expressions in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent specific mathematical relationships. These expressions can range from simple binomials, such as \( 9x^{2}-100 \), to more intricate polynomials with multiple terms.

The key to working with algebraic expressions is understanding how to manipulate them using algebraic rules and identities. This includes operations like addition, subtraction, multiplication, division, and factoring. Factoring, as demonstrated in our exercise, is the process of breaking down an expression into simpler, multiplicative components. Mastery over the manipulation of algebraic expressions is vital for solving equations and understanding higher-level math concepts.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the simple manipulation of algebraic expressions and equations. It serves as the foundation for all subsequent algebraic studies, including advanced courses like calculus and linear algebra.

Concepts such as the difference of squares, perfect square trinomials, and the general understanding of algebraic expressions form the bedrock of elementary algebra. These building blocks enable students to solve for unknown variables, simplify expressions, and ultimately, to model and solve real-world problems. The exercise provided, factoring the binomial \( 9x^{2}-100 \), offers a glimpse into the practical application of these key concepts within elementary algebra.