Problem 26 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &3 x^{2}-11 x+6\\\ &3 x^{2}+4 x-4 \end{aligned}$$

Step-by-Step Solution

Answer

The Lowest Common Multiple (LCM) of the given polynomials is \(3x^{3}-6x^{2}-13x+12\).

Step 1: Factoring the polynomials

Begin by factoring each of the two given polynomials. For \(3x^{2}-11x+6\), the factors are \(3x-2\) and \(x-3\). For \(3x^{2}+4x-4\), the factors are \(3x-2\) and \(x+2\).

Step 2: Applying the LCM rule

The LCM of polynomials is found by including the highest power of each factor present in the factored forms of the polynomials. From step one, the factors were \(3x-2\), \(x-3\) and \(x+2\). So, the LCM will be the product of these factors, which is \((3x-2)(x-3)(x+2)\).

Step 3: Simplifying to get the LCM

Finally, expand the product from step 2 to get the LCM of the two polynomials. The final LCM, after the expansion, is \(3x^{3}-6x^{2}-13x+12\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler factors that can be multiplied together to obtain the original polynomial. The goal is to express the polynomial as a product of its factors, much like factoring numbers into their prime components. This technique is particularly useful when trying to simplify polynomials or solve polynomial equations.

For instance, consider the polynomial \(3x^{2} - 11x + 6\). The process of factoring would involve finding two binomials whose product yields the original polynomial. In this case, the factors are \((3x - 2)\) and \((x - 3)\). This means that when these two binomial expressions are multiplied, the result is the original quadratic polynomial.

Similarly, for the polynomial \(3x^{2} + 4x - 4\), it can be factored into \((3x - 2)\) and \((x + 2)\). Factoring is a critical step in finding the least common multiple (LCM) of polynomials because it reveals the component expressions that must be considered when determining the LCM.

Polynomial Expressions

Polynomial expressions are mathematical phrases composed of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in algebra and appear in various equations across many math-related fields.

The polynomial expression \(3x^{2} - 11x + 6\) is a quadratic polynomial, meaning it has a degree of 2 because the highest exponent of \(x\) is 2. Similarly, \(3x^{2} + 4x - 4\) is another quadratic polynomial. These expressions are not just a random collection of numbers and variables but follow specific rules that allow us to manipulate and simplify them.

• The degree of a polynomial is the highest exponent in the expression, indicating the polynomial's highest power.
• Each term in a polynomial consists of a coefficient (a number) and a variable raised to an exponent.
• Understanding how to manage polynomial expressions helps in performing operations like addition, subtraction, multiplication, and factorization.

Recognizing the structure of polynomial expressions is crucial for applying various algebraic operations necessary for solving problems.

Algebraic Multiplication

Algebraic multiplication refers to the process of multiplying algebraic expressions, each consisting of numbers and variables. This operation is essential in handling polynomials, where each term within the polynomials needs to be multiplied together to simplify or expand expressions.

In the context of finding the least common multiple (LCM) of polynomials, algebraic multiplication plays a key role. Once the given polynomials are factored, calculating the LCM involves multiplying these factors while considering different powers or occurrences of each factor.

For example, when we take the factors \((3x - 2)\), \((x - 3)\), and \((x + 2)\) from our original polynomials, we multiply them together:

• First, multiply \((3x - 2)\) and \((x - 3)\) to get a resultant polynomial.
• Then, multiply that result by \((x + 2)\) to finalize the LCM.

The result is a new polynomial expression, in this case, \(3x^{3} - 6x^{2} - 13x + 12\), which is the LCM. This entire process relies heavily on accurate algebraic multiplication to ensure the correct simplification and expansion of expressions.