Problem 35 - 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. $$10 y^{2}+43 y-9$$

Step-by-Step Solution

Answer

\(10y^{2}+43y-9\) factors into \((5y-1)(2y+9)\).

Step 1: Factoring

Attempt to factor the trinomial. Look for a pair of integers that multiply to -90 (product of a and c in the trinomial, 10*-9) and add to 43 (b in the trinomial). The integers satisfying these conditions are 45 and -2. Therefore, the trinomial can be factored as follows: \(10y^{2}+43y-9\) = \(5y-1\)(\(2y+9\).

Step 2: FOIL check

Use the FOIL method to expand these binomials and verify the correctness of the factorization. \n FOIL the factors: \n First: \(5y * 2y = 10y^{2}\), \n Outer: \(5y * 9 = 45y\), \n Inner: \(-1 * 2y = -2y\), \n Last: \(-1 * 9 = -9\). Add all terms: \(10y^{2} + 45y - 2y - 9 = 10y^{2} + 43y - 9\). The result matches the original trinomial, so the factorization is correct.

Step 3: Answer

Conclude that the trinomial \(10y^{2} + 43y - 9\) factors into \((5y-1)(2y+9)\).

Key Concepts

FOIL method

The FOIL method is a handy tool used in algebra to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, referring to the order in which you multiply terms.
This technique helps to simplify the process of expanding expressions, ensuring you don't miss any terms. When given two binomials, such as \(a + b\) and \(c + d\), the method works as follows:

• First: Multiply the first terms in 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 in each, \(b \times d\).

After calculating each of these components, add them together: \(ac + ad + bc + bd\).
This method is particularly useful for checking the factorization of trinomials. It ensures that when you expand the factors, you return to the original trinomial expression, confirming the accuracy of the factorization.

Polynomials

Polynomials are expressions that consist of variables, coefficients, and exponents, all combined using addition, subtraction, and multiplication. A basic polynomial might look like \(3x^2 + 4x - 7\). The term "polynomial" can describe a single term, such as \(7x\), or multiple terms combined.
Each part of a polynomial is referred to as a "term," where a term is composed of a coefficient (a number) multiplying a variable raised to an exponent. The degree of a polynomial is the highest power of the variable present in the expression.
Polynomials play a crucial role in algebra, from simplifying expressions to solving equations. They can be classified based on the number of terms they have:

• Monomial: A single term (e.g., \(4x\)).
• Binomial: Two terms (e.g., \(3x + 5\)).
• Trinomial: Three terms (e.g., \(x^2 + 4x + 4\)).

Factoring trinomials, in particular, involves expressing them as the product of binomials, a skill that is essential in solving quadratic equations and simplifying higher-degree polynomials.

Algebraic expressions

Algebraic expressions are sentences in the mathematical language, combining numbers and letters, often representing unknown values called variables. In expressions like \(2x + 5\), \(x\) is the variable, while 2 and 5 are constants.
Expressing relationships and forming equations are part of the core purpose of algebraic expressions. They can represent real-world quantities and situations, providing the foundation for more complex mathematical computations.
It's important to become familiar with how to manipulate algebraic expressions. Key operations include:

• Simplification: Combining like terms to reduce the expression to its simplest form.
• Evaluation: Substituting a particular value for a variable to calculate the expression.
• Factoring: Rewriting expressions as a product of simpler expressions, often used to simplify polynomials.

Understanding algebraic expressions and their properties is crucial, as they form the basis of solving equations and inequalities, helping resolve both simple and complex algebraic problems.