Problem 29 - College Algebra

Question

Find each product. $$\left(7 x^{2}-2\right)\left(3 x^{2}-5\right)$$

Step-by-Step Solution

Answer

The result of the multiplication of the two binomials is \(21x^{4}-41x^{2}+10\).

Step 1: Expand using the FOIL Method

Apply the First, Outer, Inner, Last (FOIL) method to expand (multiply out) the two binomials. Multiply the first terms from each binomial together: \(7x^{2} * 3x^{2} = 21x^{4}\). Multiply the outer terms: \(7x^{2} * -5 = -35x^{2}\). Multiply the inner terms: \(-2 * 3x^{2} = -6x^{2}\). And multiply the last terms: \(-2 * -5 = 10\).

Step 2: Combine Like Terms

Combine the similar terms obtained from the previous multiplication, these terms are \(-35x^{2}\) and \(-6x^{2}\). However, in this case, they get combined to form \(-41x^{2}\)

Step 3: Write the Final Answer

Finally, rewrite the answer combining all the terms obtained from the previous steps, which results in the final answer: \(21x^{4}-41x^{2}+10\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a critical component in algebra that allows students to simplify expressions and solve more complex equations. It involves multiplying each term in one polynomial by each term in the other polynomial. To master this concept involves understanding what a term is—a combination of numbers and variables (such as \(3x^2\) or \(7x\)) that are multiplied together—and how to properly combine them when they are multiplied across polynomials.

Although at first glance it might seem daunting, the process is just an extension of basic distributive multiplication. When multiplying polynomials, it's important to keep track of every individual multiplication before combining like terms. This careful approach ensures that all possible products are accounted for and leads to accurate results.

Combining Like Terms

Once we've multiplied the relevant terms, we often find ourselves with an expression that contains 'like terms'. These are terms in a polynomial that have exactly the same variable factors, which means they have the same variable raised to the same power. Like terms can be combined together to simplify an algebraic expression and is a fundamental step in polynomial multiplication.

For instance, if we end up with terms such as \( -35x^2\) and \( -6x^2\), like in our original exercise, they are considered like terms because the variable part, \(x^2\), is identical. Combining like terms is straightforward: just add or subtract the numerical coefficients, depending on their sign. Thus, \( -35x^2\) and \( -6x^2\) combine to give \( -41x^2\), simplifying our expression and bringing us a step closer to the final answer.

Binomial Products

The FOIL method specifically applies to binomial products, where we're dealing with polynomials that have exactly two terms. Binomials, like \(7x^2 - 2\) and \(3x^2 - 5\), are multiplied together using the FOIL acronym: First, Outer, Inner, Last. This simplifies the process by breaking it into smaller, more manageable steps. First, multiply the 'first' terms of each binomial, then the 'outer' terms, followed by the 'inner' terms, and finally, the 'last' terms.

After applying the FOIL method and combining like terms, students can achieve a clear and simplified answer to a problem that might have initially seemed complex. Understanding how to apply this methodology to binomial products is vital for a strong foundation in algebra and will be frequently revisited in a variety of mathematical contexts.