How to Factorize Quadratic Expressions in Algebra

Factorization is a fundamental skill in algebra that involves expressing a polynomial as a product of its factors. This process simplifies the polynomial, making it easier to solve equations or analyze the polynomial's behavior. Let's explore how to factorize the expression 2x^2 - 11x^3 5x^4.

Step-by-Step Factorization

Consider the expression 2x^2 - 11x^3 5x^4. The first step is to identify the common factor in all terms. In this case, the common factor is x^2.

Step 1: Identify the Common Factor

To identify the common factor, we look for the greatest power of x that divides all terms. Here, x^2 is the common factor.

Step 2: Factor Out the Common Factor

Once we have identified the common factor, we factor it out:

2x^2 - 11x^3 5x^4 x^2(2 - 11x 5x^2)

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression inside the parentheses: 2 - 11x 5x^2. To do this, we look for two numbers that multiply to the constant term (2) and add up to the coefficient of the linear term (-11). The numbers -1 and -10 meet these criteria.

Now, we can use these numbers to rewrite the middle term and factor the expression:

2 - 11x 5x^2 2 - 1 - x 5x^2 2(1 - 5x) - x(1 - 5x) (1 - 5x)(2 - x)

Final Answer

Putting it all together, we have:

2x^2 - 11x^3 5x^4 x^2(2 - 11x 5x^2) x^2(1 - 5x)(2 - x)

Summary

In summary, when factorizing 2x^2 - 11x^3 5x^4, we identified the common factor x^2, factored it out, and then factored the resulting quadratic expression. The final factorization is:

2x^2 - 11x^3 5x^4 x^2(1 - 5x)(2 - x)

Practice and Further Resources

This process is a common method for factorizing quadratic expressions. To further enhance your skills, practice with similar problems or check out online resources and textbooks that provide more detailed explanations and examples.