Problem 52 - 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. $$15 x^{2}-31 x y+10 y^{2}$$

Step-by-Step Solution

Answer

The factorized form of \(15x^{2}-31 x y+10 y^{2}\) is \((5x - 2y)(3x - 5y)\)

Step 1: Factoring the Trinomial

Choose two binomials such that the product of the first terms equals \(15x^{2}\), the cross product equates to \(-31xy\) and product of the last terms equals \(10y^{2}\). This implies our binomials are \((5x - 2y)\) and \((3x - 5y)\). Therefore, \[15 x^{2}-31 x y+10 y^{2}= (5x - 2y)(3x - 5y)\]

Step 2: Checking the Factorization with FOIL

To confirm, use FOIL which stands for First, Outer, Inner and Last. \[First: (5x)(3x) = 15x^{2}\] \[Outer: (5x)(-5y) = -25xy\] \[Inner: (-2y)(3x) = -6xy\] \[Last: (-2y)(-5y) = 10y^{2}\] Adding these up: \[15x^{2} - 25xy - 6xy + 10y^{2} = 15x^{2} - 31xy + 10y^{2}\] This is same as our original expression confirming the factoring is correct.

Key Concepts

FOIL Method

The FOIL method is a technique used in algebra to multiply two binomials. The acronym stands for First, Outer, Inner, and Last, representing the order in which you multiply the terms of the binomials. In a factoring context, it helps to check whether the product of two binomials correctly expands to a given trinomial.

To apply the FOIL method, you take the first term of each binomial and multiply them (First), then multiply the outer terms (Outer), multiply the inner terms (Inner), and lastly, multiply the last terms of each binomial (Last). Once these four multiplications are complete, the results are combined to produce the expanded form. For example, to confirm the factorization of the trinomial \(15 x^{2}-31 x y+10 y^{2}\), we would multiply the binomials \((5x - 2y)\) and \((3x - 5y)\) using FOIL, yielding \(15x^{2} - 31xy + 10y^{2}\), which matches the original trinomial, thus verifying the factorization.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials to form a polynomial. Each term in the first binomial is multiplied by each term in the second binomial, where the FOIL method can be applied as a quick strategy, especially in binomial multiplication.

For example, if we consider the product of \((5x - 2y)\) and \((3x - 5y)\), we multiply the terms in pairs according to the FOIL sequence:

• \((5x)\cdot(3x)\) as the First terms,
• \((5x)\cdot(-5y)\) as the Outer terms,
• \((-2y)\cdot(3x)\) as the Inner terms,
• \((-2y)\cdot(-5y)\) as the Last terms.

The results from each multiplication are then combined, and like terms are added together, giving us the expanded form of the binomial product which, in this case, is \(15 x^{2}-31 x y+10 y^{2}\).

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division). A trinomial is a specific type of algebraic expression that contains three terms, typically in the form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable.

The process of factoring trinomials often involves finding two binomials whose product results in the original trinomial. This process is crucial for solving equations and simplifying expressions. When given the trinomial \(15 x^{2}-31 x y+10 y^{2}\), the challenge is to decompose it into binomials that, when multiplied, reproduce the original expression. The correct factorization, as mentioned, is \((5x - 2y)\) and \((3x - 5y)\), which simplifies the expression and allows for further algebraic manipulation such as solving for variables or evaluating the expression for specific values.