Problem 38 - Algebra 2

Question

Find the LCM of each set of polynomials. $$ 2 t^{2}+t-3,2 t^{2}+5 t+3 $$

Step-by-Step Solution

Answer

The LCM is \((t - 1)(2t + 3)(t + 1)\).

Step 1: Understand LCM of Polynomials

The Least Common Multiple (LCM) of two polynomials is the smallest polynomial (in terms of degree and coefficients) that both original polynomials divide without a remainder. This is analogous to finding the LCM of numbers.

Step 2: Factor the First Polynomial

The first polynomial is \(2t^2 + t - 3\). To factor this polynomial, we look for two numbers whose product is \(-6\) (the product of the leading coefficient and the constant term) and whose sum is \(1\) (the coefficient of the middle term). The numbers are 3 and -2. So, we split the middle term: \(2t^2 + 3t - 2t - 3\). Then we factor by grouping: \((2t^2 + 3t) - (2t + 3) = t(2t + 3) - 1(2t + 3) = (t - 1)(2t + 3)\).

Step 3: Factor the Second Polynomial

The second polynomial is \(2t^2 + 5t + 3\). We need two numbers whose product is \(6\) (the product of the leading coefficient and the constant term) and sum is \(5\). The numbers are 2 and 3. So, we split the middle term: \(2t^2 + 2t + 3t + 3\). Then we factor by grouping: \((2t^2 + 2t) + (3t + 3) = 2t(t + 1) + 3(t + 1) = (2t + 3)(t + 1)\).

Step 4: Determine the LCM

To determine the LCM, we take the highest degree of each factor present in either polynomial. The factors from the first polynomial are \((t - 1)\) and \((2t + 3)\), and from the second polynomial, the factors are \((2t + 3)\) and \((t + 1)\). The LCM is the product of the highest power of each unique factor: \((t - 1)(2t + 3)(t + 1)\).

Step 5: Simplify and Verify LCM

The LCM is given by \((t - 1)(2t + 3)(t + 1)\). Although it's possible to expand this polynomial, for the purpose of the LCM, we leave it in its factored form to confirm that each polynomial divides it without remainder. We expand if required: \((t - 1)(t + 1) = t^2 - 1\), then multiply: \((t^2 - 1)(2t + 3) = 2t^3 + 3t^2 - 2t - 3\), which confirms it encompasses both polynomials.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a number into its prime factors, but with more variables involved. When you factor a polynomial, you look for expressions that can be multiplied together to give back the polynomial you started with.

For example, consider the polynomial \(2t^2 + t - 3\). To factor it, find two numbers that multiply to give the product of the leading coefficient and the constant term. This means we're looking for numbers that multiply to \(-6\) but add up to \(1\), which are \(3\) and \(-2\). You split the middle term based on these numbers, factor by grouping, and get \((t - 1)(2t + 3)\).

Factoring helps simplify polynomials and is a crucial step in finding the least common multiple of polynomial expressions. To factor efficiently:

• Identify patterns such as simple trinomial or difference of squares.
• Look for common factors in each group.
• Use the distributive property for underlying structure recognition.

Practice will make these steps straightforward and automatic.

Polynomial Division

Polynomial division can feel a bit like long division with numbers. It’s used when factoring doesn’t lead to easily recognizable linear or simpler forms directly.

In the context of finding the least common multiple (LCM) of polynomials, division helps verify that a polynomial can divide another without leaving a remainder. It checks if the factors found (while factoring) are correct.

For the exercise, polynomial division verifies that each original polynomial factorizes neatly into the LCM without residues. Try the following:

• Set up the division like long division, writing the dividend (divided by) and divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this quotient and subtract from the dividend.
• Repeat until the remainder is smaller than the divisor. If it’s zero, one polynomial divides evenly into another.

This process underlies ensuring accuracy in the LCM and serves as a robust check against mistakes in earlier steps.

Algebraic Expressions

Algebraic expressions form the backbone of algebra itself, comprising numbers, variables, and operators combined to create expressions. Understanding them is essential for any algebraic manipulation such as addition, subtraction, or finding the LCM.

These expressions are terms grouped together where terms are algebraic units connected by plus or minus signs. For instance, in \(2t^2 + 5t + 3\), each part (\(2t^2\), \(5t\), and \(3\)) is a term, and the expression as a whole is simplified through factoring or distribution.

When dealing with polynomials:

• Recognize that each polynomial is an algebraic expression.
• Polynomials are often seen in expressions with standard operations (+, -, ×, ÷).
• Relax if they sound daunting; they are just a set of terms governed by simple arithmetic.

Working with algebraic expressions translates directly to understanding operations required in finding an LCM of polynomials. Simplifying and understanding these expressions are foundational skills in all algebraic endeavors.