Problem 27 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ \left(5 x^{2}-4\right)\left(3 x^{2}-7\right) $$

Step-by-Step Solution

Answer

The product of the two binomials \((5x^2 - 4)\) and \((3x^2 - 7)\) is \(15x^4 - 47x^2 + 28\).

Step 1: Apply the FOIL methods

We follow the FOIL method to simplify this expression: \n\nFirst: Multiply the first terms of both binomials: \(5x^2 \times 3x^2 = 15x^4\). \nOuter: Multiply the outer terms: \(5x^2 \times -7 = -35x^2\) \nInner: Multiply the inner terms: \(-4 \times 3x^2 = -12x^2\). \nLast: Multiply the last terms: \(-4 \times -7 = 28\).

Step 2: Combine like terms

Combine the like terms: \(-35x^2\) and \(-12x^2\). This becomes \(-47x^2\).

Step 3: Write down the final result

Include all terms (order of terms may vary but it's customary to start with highest degree): \(15x^4 - 47x^2 + 28\).

Key Concepts

FOIL Method

When multiplying polynomial expressions, particularly binomials, the FOIL method is a powerful technique employed to simplify the process. FOIL stands for First, Outer, Inner, Last, which represents the order in which we multiply pairs of terms from each binomial.

Let's dive into how it works with an example. Imagine you are asked to multiply \((5x^2 - 4) \times (3x^2 - 7)\).According to the FOIL method, we multiply the First terms of both polynomials, the Outer terms, the Inner terms, and lastly, the Last terms. As detailed in the step-by-step solution, the results are: \(15x^4\) (First), \(-35x^2\) (Outer), \(-12x^2\) (Inner), and \(+28\) (Last).

The FOIL method simplifies the multiplication process, ensuring you don’t miss out on any term and that every possible multiplication is accounted for. It's a handy mnemonic that helps students retain the process order, especially helpful when dealing with higher-degree or multi-term polynomials.

Combining Like Terms

After utilizing the FOIL method, the next step is 'combining like terms.' This is an essential aspect of simplifying polynomial expressions. Like terms are terms that have the exact same variable raised to the same power. In our example, \(-35x^2\) and \(-12x^2\) are like terms because both terms have the variable \(x\) squared. To combine them, simply add or subtract their coefficients, depending on their sign.

In the given problem, we combine the like terms by subtracting the coefficients: \(-35 - 12\), which equals \(-47\). Therefore, \(-35x^2 - 12x^2\) simplifies to \(-47x^2\). Combining like terms is crucial because it helps reduce the expression to its simplest form, making it easier to understand and, if necessary, to further manipulate algebraically.

Polynomial Expressions

Polynomial expressions consist of variables, exponents, and coefficients combined through addition or subtraction. Each term in a polynomial holds a degree, which is determined by the value of the exponent applied to the variable. For example, in the polynomial \(15x^4 - 47x^2 + 28\), each term represents part of what makes up the polynomial: \(15x^4\) is a fourth-degree term, \(-47x^2\) is a second-degree term, and \(28\) is a zero-degree term or a constant.

A defining feature of polynomials is that they have whole number exponents, and the exponents must not be negative. This property distinguishes them from other algebraic expressions. Understanding how polynomial expressions work is a cornerstone of algebra. They function as building blocks for more complex mathematics, including functions and calculus. Grasping the simplification of polynomial expressions via techniques like the FOIL method and combining like terms sets a solid foundation for advanced problem-solving.