Problem 9 - Algebra 2

Question

Find the least common multiple of each pair of polynomials. \(x^{2}-32 x-10\) and \(2 x+10\)

Step-by-Step Solution

Answer

The least common multiple (LCM) of the pair of polynomials \(x^{2}-32x-10\) and \(2x+10\) is \(2x(x+2)(x+5)(x-34)\).

Step 1: Factor the Polynomials

Factor the polynomials. Start with \(x^{2}-32x-10\). This can be factored as \(x-34)(x+2)\). Factoring the second polynomial, \(2x+10\), yields \(2(x+5)\).

Step 2: Determine the Shared Factors

Check the factored pairs of polynomials for any common factors. After factoring, it can be seen that there is no common factor between the pairs \(x-34, x+2\) and \(2, x+5\).

Step 3: Calculate the LCM

The LCM of the polynomials is the product of the highest powers of all the factors from both polynomials. Since there were no shared factors, the LCM is simply the product of all factors from both polynomials which are \((x-34)(x+2)(2)(x+5)\). This simplifies to \(2x(x+2)(x+5)(x-34)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential step in many algebraic operations and is crucial when finding the least common multiple (LCM) of polynomials. Factoring involves breaking down a polynomial into simpler expressions called factors, which, when multiplied together, give the original polynomial. Here’s how to factor polynomials:

• Look for a common factor in each term of the polynomial. For example, in the polynomial \(2x + 10\), both terms are divisible by 2.
• Use techniques such as grouping, using special patterns (like the difference of squares), or applying the quadratic formula for more complex expressions.

In the given exercise, the polynomial \(x^{2} - 32x - 10\) was factored into \((x-34)(x+2)\), and \(2x + 10\) was factored into \(2(x+5)\). This step is crucial because it simplifies the process of finding common factors and calculating the LCM later on.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis of algebra and can represent real-world scenarios or theoretical problems. Understanding algebraic expressions is essential for manipulating and solving equations.

In our task, the polynomials \(x^{2} - 32x - 10\) and \(2x + 10\) are both algebraic expressions. Each polynomial is a specific type of algebraic expression because it contains variables raised to integer powers. Working with these expressions means knowing how to add, subtract, multiply, and calculate their LCM.

• Algebraic expressions can include constants, like the number 10 in our example.
• Variables like \(x\) can be raised to different powers, indicating the degree of the polynomial.
• Operations within expressions define how variables and numbers interact.

Mastering these operations is key to many areas of math and science, making algebraic expressions incredibly important.

LCM Calculation

The least common multiple (LCM) of polynomials is a concept similar to that of numbers. It is the smallest expression that is divisible by the given polynomials. Finding the LCM of polynomials involves a few steps.

• First, factor each polynomial completely, as was done in the exercise.
• Next, identify the highest power of each unique factor from all polynomials.
• The LCM is the product of these highest powers.

In the exercise, there were no shared factors between \((x-34)(x+2)\) and \(2(x+5)\), so we multiply all the factors together to get the LCM, which is \(2(x-34)(x+2)(x+5)\). This method guarantees that the LCM is minimal and correctly represents the multiples of each original polynomial.

Understanding how LCM works is vital as it extends to solving equations and system problems, ensuring expressions are simultaneously divisible by given polynomials.