Problem 52 - Elementary Algebra

Question

Find the product. \((3 x-1)^{2}\).

Step-by-Step Solution

Answer

Answer: The product of the expression \((3x-1)^2\) is \(9x^2 - 6x + 1\).

Step 1: Identify the terms in the binomial

In our given binomial, \((3x-1)^2\), we have \(a=3x\) and \(b=-1\).

Step 2: Apply the expansion formula

Using the expansion formula for a square of a binomial, \((a+b)^2 = a^2 + 2ab + b^2\), we will substitute the values of \(a\) and \(b\) and expand the given binomial.

Step 3: Compute the square of 'a' and 'b'

First, find the square of \(a\) and \(b\), which are \((3x)^2\) and \((-1)^2\). We have: 1. \((3x)^2 = 9x^2\) 2. \((-1)^2 = 1\)

Step 4: Compute the product of 'a' and 'b'

Now, compute the product of \(a\) and \(b\). We have: \(2ab = 2(3x)(-1) = -6x\)

Step 5: Combine the results to find the product

Now that we have all the terms, we can combine them to get the final product: \((3x-1)^2 = 9x^2 - 6x + 1\) So the product of the given expression, \((3x-1)^2\), is \(9x^2 - 6x + 1\).

Key Concepts

Square of a Binomial

Understanding how to find the square of a binomial is a handy skill in algebra. A binomial is an algebraic expression containing two terms separated by a plus or minus sign, for example, \(a + b\) or \(a - b\). When we square a binomial, we are essentially multiplying the binomial by itself. The simple formula to remember for the square of a binomial \(a+b\) is \(a^2 + 2ab + b^2\).

Let's illustrate this with an example from the exercise, \(3x-1\)^2. Following the formula, we square the first term, \(3x\), and the second term, \(1\), then multiply \(3x\) by \(1\) and double it for the middle term. This results in the expansion \(9x^2 - 6x + 1\). Knowing this pattern helps reduce calculation errors and enhances your skills in dealing with algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and operators forming a mathematical phrase. These expressions represent quantities and are essential components of algebraic equations and functions. In our exercise, \(3x-1\) is an algebraic expression with \(3x\) being the variable part and \(1\) the constant.

Making sense of algebraic expressions demands understanding that each part has a role: coefficients, like the number 3 in \(3x\), variables like \(x\), and constants, like \(1\). When dealing with the square of an algebraic expression, you're not just squaring each term individually; you're considering the combined effects of these terms when multiplied, leading to a polynomial. Recognizing these components aids in mastering skills like simplifying, factorizing, and expanding expressions.

Polynomial Multiplication

The product of two algebraic expressions leads us to the process of polynomial multiplication. When we multiply polynomials, we're combining like terms and using distribution - also known as the FOIL method for binomials, which stands for First, Outer, Inner, Last. This is pertinent when you're squaring a binomial because you're multiplying the binomial by itself.

In our example, \(3x\) and \(1\) are considered 'terms' of our binomial. When we square the binomial, we follow the steps of polynomial multiplication, applying the distributive property twice. First, \(3x\) is multiplied by itself, then it’s multiplied by \(1\) (twice, for each term), and finally, \(1\) is squared. The product of these individual multiplications are then added together to form a polynomial, in our case, \(9x^2 - 6x + 1\). Understanding this process enhances our ability to handle more complex algebraic operations and polynomials.