Question

Find the L.CM of the given polynomials. $$ x^{2}+4 x+4, x^{3}+2 x^{2}, \quad(x+2)^{3} $$

Short answer

The LCM of the given polynomials is \ x^2 (x + 2)^3 \.

Step-by-step solution

Factorize the Polynomials

First, factorize each given polynomial. Factorizing helps to identify the common and unique factors for finding the least common multiple (LCM).1. For the polynomial \(x^2 + 4x + 4\):\[x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2\]2. For the polynomial \(x^3 + 2x^2\):\[x^3 + 2x^2 = x^2(x + 2)\]3. For the polynomial \( (x + 2)^3 \):\[ (x + 2)^3 \]

Identify the Highest Power of Each Factor

Now, identify the unique factors and select the highest powers of each factor that appear in the polynomials.- The factor \(x \) appears as \(x^2\) in the second polynomial.- The factor \(x + 2 \) appears as \( (x + 2)^3 \) in the third polynomial.

Multiply the Factors

Finally, multiply the selected highest power of each factor to determine the LCM of the given polynomials.\[\text{LCM} = x^2 \times (x + 2)^3\]

Key concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler polynomials, called factors, that when multiplied together give back the original polynomial. It helps in simplifying expressions and solving polynomial equations. For example, the given polynomial \(x^2 + 4x + 4\) can be factored into \((x + 2)^2\). Factorization makes it easier to handle polynomials in various mathematical operations such as finding the Least Common Multiple (LCM).

Least Common Multiple

The Least Common Multiple (LCM) of polynomials is the smallest polynomial that is divisible by each of the given polynomials. Finding the LCM involves identifying the highest power of each unique factor that appears in any of the given polynomials. This concept is similar to finding the LCM of integers. In the given exercise, the LCM of the polynomials \(x^2 + 4x + 4\), \(x^3 + 2x^2\), and \((x+2)^3\) is found by selecting the highest powers of the factors, which are \(x^2\) and \((x + 2)^3\). Thus, the LCM is given by \(x^2 \cdot (x + 2)^3\).

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants combined using arithmetic operations. A polynomial is a type of algebraic expression where variables are raised to non-negative integer powers. Understanding algebraic expressions is essential for factorization. For instance, the term \(x^3 + 2x^2\) in the exercise is an algebraic expression that can be factored into \(x^2(x + 2)\), simplifying our calculations. Learning to manipulate these expressions through operations like addition, multiplication, and factorization forms the foundation of algebra.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This technique is crucial for solving polynomial equations and finding polynomial LCMs. To factor a polynomial, one must recognize patterns and use methods such as factoring by grouping, using special formulas, or applying the distributive property. In the exercise, we apply these techniques:
1. \(x^2 + 4x + 4\) is factored as \((x + 2)^2\) using the perfect square trinomial method.
2. \(x^3 + 2x^2\) is factored as \(x^2(x + 2)\) through factoring out the greatest common factor (GCF).
Practicing these methods sharpens skills in algebra and prepares students for more complex mathematical challenges.