Problem 57 - 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. $$12 x^{2}-25 x y+12 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(12x^{2}-25xy+12y^{2}\) is \((4x - 3y)(3x - 4y)\).

Step 1: Set Up Potential Factors

Identify numbers that multiply to 144 and add to -25. The numbers that meet these conditions are -16 and -9 because \(-16 × -9 = 144\) and \(-16 + -9 = -25\). The factorization will therefore involve these numbers and will have to be set up as \((ax - p)(bx - q)\).

Step 2: Write the Factors

Coverting -16 and -9 to the correct factors, we can now say the factors are \((4x - 3y)(3x - 4y)\) because \((4x × 3x) = 12x^{2}\) and \((3y × -4y) = 12y^{2}\). So the pattern matches our original trinomial.

Step 3: Verify Using FOIL Method

To confirm it, use the FOIL method to check. FOIL stands for First, Outer, Inner, Last, and refers to the order in which terms should be multiplied in a binomial multiplication. So, \((4x - 3y)(3x - 4y) = 12x^{2} - (16xy) - (9xy) +12y^{2} = 12x^{2} -25xy + 12y^{2}\) which is our original trinomial.

Key Concepts

FOIL Method

The FOIL Method is a technique used for multiplying two binomials. It's an essential concept in algebra, as it helps simplify the process and ensures accuracy. FOIL stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms, which are the first term of the first binomial and the second term of the second binomial.
• Inner: Multiply the inner terms, which are the second term of the first binomial and the first term of the second binomial.
• Last: Multiply the last terms of each binomial.

Calculating these products and then adding them together gives you the resulting expression. This method is crucial in verifying if the factorization of trinomials is correct, as it breaks down the process into manageable steps, making error-checking much easier.

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. They are sum of multiple monomials. Each component of a polynomial is called a "term," and these terms are separated by the '+' or '-' signs.

Key features of polynomials include:

• Degrees: The degree of a polynomial is the highest exponent of the variable. For example, in the polynomial \(12x^2 - 25xy + 12y^2\), the degree is 2.
• Coefficients: These are numbers placed in front of variables. In the same polynomial, 12, -25, and 12 are coefficients.
• Variables: Letters representing numbers, in this case, \(x\) and \(y\).

Understanding polynomials is key for mastering algebra, as they form the basis for more complex algebraic concepts. Recognizing the structure of a polynomial helps in tasks like factoring or performing operations like addition and subtraction.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators. They are like phrases in the language of mathematics, conveying meaning through a mix of these elements.

Characteristics of algebraic expressions include:

• They contain at least one term, which can be a lone variable like \(x\), or a combination like \(3xy\).
• They can include operations such as addition, subtraction, multiplication, and division between terms.
• They do not have an equality sign, as opposed to equations.

By understanding algebraic expressions, students can better comprehend how to manipulate them in solving equations or simplifying terms. Recognizing expressions as a collection of terms assists in breaking them down methodically, easing the process of learning more advanced algebraic concepts.