Problem 50 - Prealgebra and Introductory Algebra

Question

Match the polynomials with their LCM. An LCM may be used more than once. a. \(x^{2}-4\) and \(x^{2}+3 x+2\) b. \(x+3\) and \(x^{2}+5 x+6\) c. \(x^{2}-x-2\) and \(x^{2}+2 x+1\) d. \(x-4\) and \(x^{2}-1\) e. \(2-x\) and \(x^{2}+3 x+2\) f. \(4-x\) and \(x^{2}-1\) g. \(x-4\) and \(1-x^{2}\) h. \(2+x-x^{2}\) and \((x+1)^{2}\) i. \(\quad(x+3)(x+2)\) ii. \(\quad(x-4)(x+1)(x-1)\) iii. \((x+2)(x-2)(x+1)\) iv. \((x-2)(x+1)(x+1)\)

Step-by-Step Solution

Answer

The matching of polynomial pairs with their LCM is as follows: a-iii, b-i, c-iv, d-ii, e-i, f-ii, g-ii, h-iv.

Step 1 - Factoring the polynomials

Factorize the given polynomials in pairs. For example, the pair \(x^{2}-4\) and \(x^{2}+3 x+2\) can be factored as \((x-2)(x+2)\) and \((x+1)(x+2)\) respectively.

Step 2 - Finding the LCM for each pair

The LCM of the pair \((x-2)(x+2)\) and \((x+1)(x+2)\) is \( (x-2)(x+2)(x+1)\) which is option iii. The LCM is the polynomial product that includes all distinct factors of both polynomials, each to the highest power that occurs in either polynomial.

Step 3 - Repeat for all pairs

Repeat steps 1 and 2 for all pairs of polynomials.

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is one of the bedrock skills in algebra. Factoring involves expressing a polynomial as the product of its simplest polynomials, known as factors. When you encounter a polynomial like x^2 - 4, factoring it reveals (x - 2)(x + 2). This is because x^2 - 4 is a difference of squares—a common type of polynomial that can be factored into the product of a sum and difference.

Factoring polynomials requires knowledge of various methods, such as taking out a common factor, using special products formulas, or applying more advanced techniques like grouping or the quadratic formula. One tip to help break down these problems is to always look for the greatest common factor (GCF) first. If a polynomial doesn't factor neatly, it's known as prime or irreducible. An integral part of working with these expressions is recognizing patterns and applying the appropriate factoring rule.

For instance, in our example, the polynomial x^2 + 3x + 2 is easily factored into (x + 1)(x + 2) because it is a simple quadratic trinomial, where you need to find two numbers that multiply to give the last term (+2) and add up to the middle coefficient (+3).

Least Common Multiple of Polynomials

When dealing with the Least Common Multiple (LCM) of polynomials, we are searching for the smallest polynomial that each of the given polynomials can divide into without leaving a remainder. It's the equivalent of finding the LCM of integers but with algebraic expressions.

To find the LCM of two polynomials, you must first factor them completely. Then, the LCM is constructed by multiplying each different factor the greatest number of times it occurs in any of the expressions. For example, if you have the factored polynomials (x - 2)(x + 2) and (x + 1)(x + 2), the highest power of (x - 2) and (x + 1) is 1 in either polynomial, and the highest power of (x + 2) is also 1.

So, their LCM is the multiplication of all these factors, (x - 2)(x + 2)(x + 1). When two polynomials do not share common factors, their LCM is simply their product. Remember the key to finding the LCM is to make sure that the polynomial you get can be divided by each of the polynomials you started with, which is why factoring is an essential first step.

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like +, -, *, and /). These expressions represent quantities and are fundamental in formulating relationships in algebra. They can be as simple as a single term, like x, or as complex as a polynomial, such as 5x^3 - 2x^2 + x - 7.

When evaluating and simplifying algebraic expressions, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Beyond this rule, when working with algebraic expressions, especially when solving equations, combining like terms and using inverse operations to isolate variables are key strategies.

Whether you are adding, subtracting, multiplying, or finding the LCM of these expressions, understanding their structure and how they can be manipulated is crucial. Always keep in mind that the goal is to simplify the expression to its most basic form, which often unveils more about the relationship the expression is meant to describe.