Problem 75 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Find the least common multiple of the expressions. \(x^2+8 x+12, x+6\)

Step-by-Step Solution

Answer

The least common multiple of \(x^2+8x+12\) and \(x+6\) is \((x+2)(x+6)\).

Step 1: Factorize the expressions

The first step is to factorize the given expressions if possible. The expression \(x^2+8x+12\) can be factorized into \((x+2)(x+6)\). The second expression \(x+6\) is already in its simplest form.

Step 2: Determine the LCM of factorized expressions

The least common multiple (LCM) of the two factorized expressions \((x+2)(x+6)\) and \(x+6\) is obtained by multiplying each factor the greatest number of times it occurs in either expression. Here, \(x+6\) is common in both expressions, and \(x+2\) is present once in the first expression, so the LCM is \((x+2)(x+6)\).

Key Concepts

Factoring Expressions

Factoring expressions is a crucial step in solving many algebraic problems, including finding the least common multiple (LCM) of polynomial expressions. When you factor an expression, you are essentially breaking it down into smaller, simpler expressions that, when multiplied together, give you the original expression. For instance, the expression \(x^2+8x+12\) can be factored as \((x+2)(x+6)\). This means that if you multiply \((x+2)\) by \((x+6)\), you will obtain the original expression \(x^2+8x+12\).

• Factoring makes it easier to perform operations such as finding the LCM.
• It simplifies expressions, making them more manageable.
• Common methods include factoring by grouping and using special product formulas.

Let's look at an example: \(x^2+8x+12\). Here, we look for two numbers that multiply to 12 and add up to 8, which are 2 and 6. Hence, the factorization becomes \((x+2)(x+6)\).
This helps us see each component and understand its multiplicative structure.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols to solve problems. It serves as a unifying thread of almost all of mathematics, allowing us to express relationships and changes in abstract ways. In our particular exercise of finding the LCM, algebra plays a significant role as it involves manipulating algebraic expressions to find common multiples.

• Algebraic manipulation includes operations such as addition, subtraction, multiplication, and division performed among variables and constants.
• It allows us to solve equations and systems of equations.
• We use it to express patterns, sequences, and relationships between quantities.

In our example, we have to manage the expression \((x^2+8x+12)\) and work with its factored form \((x+2)(x+6)\) to simplify the problem of finding the LCM. This highlights how algebra provides the techniques necessary to deal with such expressions efficiently.

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Each term in a polynomial consists of a coefficient and a variable raised to a whole number exponent. The polynomials used in algebraic operations are typically simple to complex expressions based on the number and degree of their terms.

• The degree of a polynomial is the highest exponent of the variable in the expression.
• When dealing with polynomials, we often apply operations such as factoring to simplify and find solutions to expressions.
• Understanding polynomials helps in solving equations that model real-world phenomena.

Take for example the polynomial \(x^2 + 8x + 12\), which is a second-degree polynomial because the highest exponent of \(x\) is 2. Factoring it into \((x+2)(x+6)\) helps us break down the problem of finding common multiples, showing that factoring plays an essential role in manipulating polynomials.