Problem 36 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. \(x^{2}+6 x y+8 y^{2}\)

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}+6xy+8y^{2}\) is \((x + 2y)(x + 4y)\).

Step 1: Identify the Trinomial

The trinomial given is \(x^{2}+6 xy+8 y^{2}\). Identify the terms as \(a x^2+ bx + c\) where a=1, b=6 and c=8.

Step 2: Factor the Trinomial

Factor the trinomial by finding two numbers that multiply to give the product ac (or c if a is 1), and add up to b. In this case, those numbers are 2 and 4. Hence the factored form of the trinomial is \((x + 2y)(x + 4y)\).

Step 3: Check the Factorization using FOIL

Use the FOIL method (which stands for First, Outer, Inner, Last, a strategy used to multiply binomials) to check the factored form:\n First terms: \(x \cdot x = x^2\),\nOuter terms: \(x \cdot 4y = 4xy\),\nInner terms: \(2y \cdot x = 2xy\),\nLast terms: \(2y \cdot 4y = 8y^2\).\nSo, \(x^2 + 4xy + 2xy + 8y^2 = x^2 + 6xy + 8y^2\), which confirms that the trinomial was factored correctly.

Key Concepts

FOIL Method

Let's dive into the FOIL Method, a popular technique for multiplying two binomials. It's called "FOIL" because it helps you remember the order in which you should multiply the terms:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

For example, if we have two binomials like \((x + a)(x + b)\),in the FOIL method, we do the following:- Multiply the first terms: \(x \cdot x = x^2\).- Multiply the outer terms: \(x \cdot b = bx\).- Multiply the inner terms: \(a \cdot x = ax\).- Multiply the last terms: \(a \cdot b = ab\).
When you add everything together, you get a new polynomial: \(x^2 + bx + ax + ab\).
This technique not only helps in confirming factorizations, but also ensures that you've multiplied every part of the binomials correctly.

Polynomial Factoring

Polynomial factoring is an essential algebraic skill that simplifies expressions and solves equations. When you factor a polynomial, you're essentially splitting a complicated expression into simpler parts, or factors, that when multiplied together return the original polynomial. For example, trinomial factoring focuses on expressions with three terms.

• Look for patterns like \(ax^2 + bx + c\) where \(a = 1\).
• Identify two numbers that multiply to provide \(c\)and add to give \(b\).
• Use these numbers to rewrite the middle term for easier factoring.

By breaking down the expression in this way, you simplify solving equations and can quickly find roots or zeros of the polynomial. It lends a structure to algebraic expressions and opens up more avenues for solving.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations (like addition and multiplication) that represent mathematical relationships. In expressions like \(x^2 + 6xy + 8y^2\), we see:

• Constants: Numbers like 8, which are fixed values.
• Variables: Symbols, typically \(x\) or \(y\), that represent unknown values.
• Coefficients: Numbers like 6 that are multiplied by the variables.

Algebraic expressions differ from equations as they don't have an "equals" sign providing a specific value. They're essential for representing patterns and formulating problems in algebra. Understanding these expressions is crucial. It lays the groundwork for more complex concepts, such as functions and calculus.
By mastering algebraic expressions, students develop a toolkit for solving a wide variety of mathematical challenges.