Problem 45 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}+5 x y+2 y^{2}$$

Step-by-Step Solution

Answer

The factorization of the given trinomial \(3x^{2}+5xy+2y^{2}\) is \((3x+2y)(x+y)\). This has been confirmed using the FOIL multiplication method.

Step 1: Factoring the Trinomial

When factoring a trinomial of the form \(ax^{2}+bxy+cy^{2}\), first look for common factors. In this case, there are no common factors. Then look for two numbers that multiply to \(ac = 3*2 = 6\) and add to \(b = 5\). The numbers that fit this criteria are 3 and 2. So, the factorisation will be \((3x+2y)(x+y)\).

Step 2: Using FOIL to Check the Factorization

We can verify the factorisation by using the FOIL method, which stands for First, Outer, Inner and Last. Multiply the first terms in each binomial, then the outer terms, then the inner terms, and finally the last terms, and add up the results:First terms: \(3x * x = 3x^{2}\)Outer terms: \(3x * y = 3xy\)Inner terms: \(2y * x = 2xy\)Last terms: \(2y * y = 2y^{2}\)Adding these up gives the original trinomial: \(3x^{2} + 5xy + 2y^{2}\). So, the factoring has been verified.

Key Concepts

FOIL Method

The FOIL method is an acronym that stands for First, Outer, Inner, and Last. It's a handy mnemonic that helps you remember how to multiply two binomials. It's particularly useful in polynomial factorization, crucial for simplifying and solving algebraic expressions.

When you apply the FOIL method, you're taking the first terms of each binomial and multiplying them together, followed by the outer terms, the inner terms, and finally, the last terms. Here’s a breakdown:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

After multiplying, you add up all the resulting values to get a single polynomial expression. The FOIL method is especially important when factoring trinomials because it provides a way to check your work. By reversing the process, you can multiply the factors to ensure you have the correct original polynomial.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. Factors are simpler polynomials whose product equals the original polynomial. Factoring is a fundamental tool in algebra because it simplifies expressions and is necessary for solving equations.

For a trinomial of the form \(ax^{2}+bxy+cy^{2}\), the factorization process involves finding two binomials that, when multiplied using the FOIL method, give you the original trinomial.

Looking for Patterns

In the factoring process, it's important to look for patterns like the difference of squares or perfect square trinomials. These patterns can give you a hint about which factors are involved.

Searching for Factors

For trinomials, you specifically want to find two numbers that multiply to the product of the quadratic term's coefficient and the constant term, and sum to the coefficient of the middle term. This step can sometimes require a bit of trial and error.

Once you find a successful factorization, it's crucial to check your work. The FOIL method comes in handy here to ensure that the product of the factors indeed equals the original trinomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are the building blocks of algebra and can represent real-world scenarios quantitatively.

An example of an algebraic expression is a quadratic trinomial, which is a second-degree polynomial (the highest power of the variable is two) with three terms. As seen in the exercise \(3x^{2}+5xy+2y^{2}\), the expression includes terms of different degrees.

Understanding Terms

Each term in an algebraic expression contains a coefficient (numerical part) and often a variable part (which may include exponents). Trinomials have a particular structure that can be used to help in factorization, as demonstrated in this exercise.

Working with algebraic expressions enables us to solve equations by finding the values of the variables. Factoring plays a key role in simplification and solving since it breaks down complex expressions into simpler, more manageable pieces.