Problem 56 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (1+\sqrt{3 x})^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((1 + \sqrt{3x})^2\). Answer: The simplified expression is \(1 + 2\sqrt{3x} + 3x\).

Step 1: Apply the binomial formula

In this step, we will apply the binomial formula to expand the given expression. Recall that the binomial formula for \((a + b)^{2}\) is: $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$ Here, \(a = 1\) and \(b = \sqrt{3x}\).

Step 2: Substitute the values

Now, substitute \(a = 1\) and \(b = \sqrt{3x}\) into the binomial formula: $$(1 + \sqrt{3x})^{2} = 1^{2} + 2(1)(\sqrt{3x}) + (\sqrt{3x})^{2}$$

Step 3: Simplify the expression

In this step, we will simplify the expression obtained in Step 2: $$1^2 + 2(1)(\sqrt{3x}) + (\sqrt{3x})^2 = 1 + 2\sqrt{3x} + 3x$$ The simplified expression is: $$1 + 2\sqrt{3x} + 3x$$

Key Concepts

Binomial Formula

The binomial formula is a cornerstone of algebra that helps us expand expressions that are raised to a power. In particular, when we have a binomial—that is, an algebraic expression with two terms—and we need to raise it to the second power, the formula comes in handy to simplify the process without actually multiplying the entire expression out.
For a binomial \(a + b\), raised to the power of two, the formula is:\[ (a + b)^{2} = a^{2} + 2ab + b^{2} \]This succinctly captures the idea that when you square a binomial, you get the square of the first term, plus twice the product of the two terms, plus the square of the second term. Applying the binomial formula saves time and reduces the chance of making errors during expansion.
In educational practice, it is valuable for students to memorize this formula, not just for ease of simplification but also to develop an understanding of how polynomial expansion works. In the given exercise, \((1 + \sqrt{3x})^{2}\) illustrates the use of the binomial formula with the two terms being 1 and \(\sqrt{3x}\).

Radicals in Algebra

Radicals, or roots, are the opposite operation of an exponent and they can often appear in algebraic expressions. Simplifying expressions with radicals involves several rules, such as the product and quotient rule, as well as rationalizing the denominator if necessary.
In the context of the given exercise, the radical \(\sqrt{3x}\) is involved in the binomial expression. Working with radicals in this situation requires understanding that when you square a square root, the radical symbol is eliminated, since \(\sqrt{a}\)^2 = a. This identities in simplifying the given exercise as follows:\[ (\sqrt{3x})^2 = 3x \]Incorporating radicals into algebraic expressions necessitates a familiarity with their properties to simplify and manipulate the expressions correctly. It's important for students to understand these properties to feel confident in tackling problems involving radicals.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials as well as expansion and factoring. These operations build upon the rules of arithmetic but are applied to algebraic expressions that include variables raised to various powers.
In our specific exercise, once we apply the binomial formula, we engage in polynomial operations by collecting like terms and simplifying. For the expression \(1 + 2\sqrt{3x} + 3x\), we recognize that there are no like terms to combine aside from the ones that result directly from applying the binomial expansion. The operation here is quite straightforward as it involves adding constants and coefficients in front of radicals and variables.
A sound grasp of polynomial operations is essential in algebra because it allows students to simplify expressions, solve equations, and understand the behavior of functions. As part of the exercise improvement advice, students are encouraged to practice these operations frequently to become proficient at spotting patterns and shortcuts that can aid in the simplification of polynomials.