Problem 7 - 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. $$5 y^{2}-16 y+3$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(5y^2-16y+3\) is \((5y+3)(y-1)\)

Step 1: Factoring the trinomial

A trinomial of the form \(ay^2 + by + c\) is factored as \((dy+e)(fy+g)\) where d*f=a, e*g=c, and d*g + e*f=b. For our trinomial \(5y^2-16y+3\), we need to find two numbers that multiply to 5*3 = 15 (because 5 is the coefficient of \(y^2\) and 3 is the constant) and add up to -16 (the coefficient of y). These numbers are -5 and -3. So, we can write the trinomial as \((5y+3)(y-1)\)

Step 2: Checking the factorization using FOIL

In the FOIL method, we multiply the First terms of both binomials, then the Outer, then the Inner and finally, the Last terms. From \((5y+3)(y-1)\), \nFirst: \(5y*y = 5y^2\), \nOuter: \(5y*-1 = -5y\), \nInner: \(3*y = 3y\), \nLast: \(3*-1 = -3\). \nAdding them, we get \(5y^2-5y+3y-3 = 5y^2-16y+3\) which is our original trinomial thus verifying our factorization.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. They are the building blocks of algebra, allowing us to represent quantities and perform calculations in a flexible way.
For example, an expression like \(5xy + 3\) involves a variable \(y\), a constant \(3\), and a coefficient \(5\) which multiplies both. This expression can represent different quantities depending on the value of \(y\).
Working with algebraic expressions often involves simplifying or rearranging terms. When there are multiple terms present like in trinomials, which have three parts, factoring can simplify the expression further.
In the exercise, we dealt with a trinomial \(5y^2-16y+3\), which is a specific type of algebraic expression used frequently in algebra to solve equations.

Polynomials

Polynomials are expressions made up of variables and coefficients, where the variables are raised to whole number powers. They can range from simple expressions, like monomials, to more complex ones, such as trinomials.
In a polynomial, the highest power of the variable determines its degree. For instance, \(5y^2 - 16y + 3\) is a second-degree polynomial because the highest exponent of \(y\) is 2.
Polynomials play a crucial role in algebra because they often appear in equations that need solving. Factoring polynomials, like we did in the exercise, involves expressing the polynomial as the product of its factors, which are simpler expressions.
This helps in understanding the behavior of the polynomial, finding zeros, and solving polynomial equations more easily. Recognizing patterns in polynomials can simplify complex mathematical problems, making factoring an essential skill in algebra.

FOIL Method

The FOIL method is a handy technique used in algebra to multiply two binomials. The term 'FOIL' stands for:

• **F**irst - Multiply the first terms of each binomial
• **O**uter - Multiply the outer terms of the binomials
• **I**nner - Multiply the inner terms
• **L**ast - Multiply the last terms

This method helps ensure that you account for all interactions between terms when expanding expressions like \((5y + 3)(y - 1)\). This is precisely what we used in our step-by-step solution to verify our factorization.

Following the steps:
* First: \(5y \cdot y = 5y^2\)
* Outer: \(5y \cdot -1 = -5y\)
* Inner: \(3 \cdot y = 3y\)
* Last: \(3 \cdot -1 = -3\)
Combining these results gives us the original trinomial \(5y^2 - 16y + 3\), confirming that our factorization in the exercise is correct.
The FOIL method is an essential tool that adds accuracy to polynomial multiplication and ensures all terms are accounted for accurately.