Problem 46 - 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. $$3 x^{2}+11 x y+6 y^{2}$$

Step-by-Step Solution

Answer

The trinomial \(3x^2+11xy+6y^2\) factors into \((3x+2y)(x+3y)\). And it's verified by using the FOIL method.

Step 1: Identifying The Type of Trinomial

The major part of solving this problem is to identify the trinomial type. In this case, it is a trinomial in the form of \(ax^2 + bxy + cy^2\), where a, b, and c are coefficients and x, y are variables.

Step 2: Factoring The Trinomial

We have to factorize the trinomial \(3x^2+11 xy+6 y^2\). This can be done by finding two numbers that multiply to give \(ac = 3*6 =18\), and add up to \(b = 11\). The two numbers satisfying these conditions are 2 and 9. We then rewrite the middle term \(11xy\) as \((2xy + 9xy)\), giving us \(3x^2+2xy+9xy+6y^2\). Grouping yields \((3x^2+2xy)+(9xy+6y^2)\), which simplifies into \(x(3x+2y)+3y(3x+2y)\) and finally \((3x+2y)(x+3y)\). Thus, the factorization of the trinomial is \((3x+2y)(x+3y)\).

Step 3: Verification Through FOIL Multiplication

To verify the solution, use the FOIL method to expand the factorized form. FOIL stands for 'First', 'Outer', 'Inner', and 'Last', representing the terms to multiply together. \n- 'First' terms: \(3x * x = 3x^2\), \n- 'Outer' terms: \(3x * 3y = 9xy\), \n- 'Inner' terms: \(2y * x = 2xy\), \n- 'Last' terms: \(2y * 3y = 6y^2\). \nAdding them gives \(3x^2+9xy+2xy+6y^2 = 3x^2+11xy+6y^2\), which verifies our factored form.

Key Concepts

FOIL Multiplication

Understanding FOIL multiplication is essential in algebra, particularly when dealing with polynomials. FOIL is an acronym that stands for First, Outer, Inner, and Last. It's a method used for multiplying two binomials.

For instance, suppose you have \( (a+b) \times (c+d) \). Here's how you use FOIL:

• 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: \( b \times d \).

After multiplying, you add the results to get the expanded form of the binomial multiplication. This method is particularly helpful when verifying the factorization of trinomials, as seen in the given exercise.

Polynomial Factorization

Polynomial factorization is a critical tool in algebra that breaks down complex expressions into simpler, multiplied factors. This simplification can help solve equations, graph functions, and understand polynomial behavior. When factoring trinomials, like \( ax^2 + bxy + cy^2 \), the goal is to find two binomials that, when multiplied using the FOIL method, give back the original trinomial.

To factor such a trinomial, look for two numbers that multiply to the product of the coefficient of \( x^2 \) and the constant (ac) and also add up to the coefficient of the middle term (b). Rewriting the middle term with these two numbers allows you to group and factor by grouping, simplifying to a product of binomials, as seen in the step-by-step solution above.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Expressions can range from simple, like \( 2x+3 \), to complex, like the trinomial \( 3x^2+11xy+6y^2 \) in the given exercise. These expressions do not have an equal sign, unlike algebraic equations.

An understanding of how to work with algebraic expressions is fundamental to algebra. This includes operations such as addition, subtraction, multiplication, division, and factoring. For example, factoring is a way to transform a complicated expression into a product of simpler expressions, which sometimes reveals solutions to equations or simplifies calculations.