Problem 48 - College Algebra

Question

Find each product. $$\left(5 x^{2}-3\right)^{2}$$

Step-by-Step Solution

Answer

The product of \((5x^2 - 3)^2\) is \(25x^4 - 30x^2 + 9\).

Step 1: Identifying the binomial

The given expression is \((5x^2-3)^2\). Here the binomial is \(5x^2 - 3\), Written in the form \(a - b\), we can identify \(a = 5x^2\) and \(b = 3\).

Step 2: Applying the square of a binomial formula

Now we apply the formula \((a - b)^2 = a^2 - 2ab + b^2\) to our binomial \((a = 5x^2, b = 3)\). Therefore, \((5x^2 - 3)^2 = (5x^2)^2 - 2*(5x^2)*3 + 3^2\).

Step 3: Simplifying the solution

Simplify each term to get: \(25x^4 - 30x^2 + 9\).

Key Concepts

Binomial Expressions

When we refer to binomial expressions, we're talking about algebraic expressions that contain exactly two terms, which are typically separated by a plus (+) or minus (-) sign. These terms can be numbers, variables, or a combination of both. For example, in the product ewline ewline ewline ewline ewline ewline ewline (5x^2−3)^2ewline ewline ewline ewline , the expression inside the parentheses, ewline ewline ewline 5x^2−3ewline ewline ewline ewline , is a binomial with 5x^2 as the first term and -3 as the second term.To make the concept of binomials even clearer, let's consider that each term can have coefficients (like the 5 in 5x^2), variables (like x in 5x^2), and exponents (like the 2 in x^2), and they are extremely useful in various branches of mathematics including algebra.

Algebraic Formulas

Moving on to algebraic formulas, these are the tools that allow us to simplify expressions, solve equations, and understand mathematical relationships. One fundamental algebraic formula is the square of a binomial, which takes the form ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline (a±b)^2 = a^2 ± 2ab + b^2ewline ewline ewline ewline . This particular formula helps us expand binomials that are being squared without having to multiply the binomial by itself in a long-form.By memorizing and applying these formulas, students can quickly and accurately expand binomials, factor expressions, and even graph equations. It's a staple for algebra students to learn these formulas as they form the foundation for more advanced concepts in algebra and calculus.

Polynomial Expansion

Lastly, we touch upon the concept of polynomial expansion. A polynomial is an expression that can have constants, variables, and exponents, that are combined using addition, subtraction, multiplication, and non-negative integer exponents. When we expand a polynomial, we simplify the expression by multiplying and combining like terms wherever possible.For example, when expanding the square of a binomial, we're actually performing polynomial expansion. Taking the previous exercise, ewline ewline ewline (5x^2 - 3)^2ewline ewline ewline ewline , by applying our algebraic formula for the square of a binomial, we get an expanded polynomial ewline ewline ewline 25x^4 - 30x^2 + 9ewline ewline ewline ewline . Understanding this process is critical for students as it appears in many algebraic operations and is a precursor to understanding more complex functions and calculus.