Problem 4 - 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. $$2 x^{2}+7 x+3$$

Step-by-Step Solution

Answer

The factors of given trinomial \(2x^{2}+7x+3\) are \( (2x + 1)\) and \( (x + 3)\)

Step 1: Factor the Trinomial

Factor the trinomial \(2x^{2}+7x+3\). Look for two numbers which when added give 7 and when multiplied give 6 (which is the product of 2 and 3). These numbers are 1 and 6. We write the trinomial as follows: \(2x^{2}+x+6x+3\) and then factor out common factors from each pair of terms: \(x(2x+1)+3(2x+1)\). This gives \( (2x + 1)(x + 3)\), which are the factors.

Step 2: Verify using FOIL

Use the FOIL (First, Outer, Inner, Last) to multiply the factors and check if you get the original trinomial. ((2x * x) + (2x * 3) + (1 * x) + (1 * 3)) simplifies to \(2x^{2}+7x+3\). We can see that it is indeed same as the original trinomial, confirming that the factoring is correct.

Key Concepts

FOIL Method

The FOIL method is a technique used in algebra to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in the binomials.

For example, to apply the FOIL method for \( (a + b)(c + d) \), you would multiply the terms in this order:

• First: Multiply the first terms of each binomial: \( a \times c \).
• Outer: Multiply the outer terms: \( a \times d \).
• Inner: Multiply the inner terms: \( b \times c \).
• Last: Multiply the last terms of each binomial: \( b \times d \).

After multiplying, you would then combine like terms to simplify the expression.

In the exercise \(2x^{2}+7x+3\), after factoring, we get two binomials \( (2x + 1)(x + 3) \). To check the factorization using FOIL, we multiply these binomials:

• First: \(2x \times x = 2x^2\)
• Outer: \(2x \times 3 = 6x\)
• Inner: \(1 \times x = x\)
• Last: \(1 \times 3 = 3\)

Then we combine the middle terms (6x and x) to get \(2x^{2}+7x+3\), which matches the original trinomial, thus confirming the correct factorization.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of its factors, much like factoring a number into its prime components. This process often simplifies solving equations and understanding properties of functions.

For instance, the trinomial \(2x^{2}+7x+3\) can be factored into \(2x^{2}+x+6x+3\), and then further into \(x(2x+1)+3(2x+1)\), as seen in the exercise. Finally, we recognize that both terms contain \(2x+1\), thus the fully factored form is \( (2x + 1)(x + 3) \).

Some tips for effective polynomial factorization include:

• Looking for a greatest common factor first.
• Using the Reverse FOIL method for quadratics where you split the middle term.
• Checking your factors by multiplying them to see if you get back the original polynomial.

Understanding how to factor polynomials is crucial for further algebraic manipulations and solving higher-degree polynomial equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They can range from simple expressions like \(3x + 2\) to more complex polynomials like the trinomial we encountered \(2x^{2}+7x+3\). Expressions become the cornerstone for forming equations and inequalities, which in turn are solved to find the values of unknown variables.

When working with algebraic expressions, it's important to understand how to perform operations like addition, subtraction, multiplication, and division, as well as more advanced manipulations such as factoring and expanding expressions. A strong grasp of these concepts allows students to tackle more complex problems in algebra and beyond.