Factoring 4x2 + 23x + 72: Simplified Expression

Quick Links :

The process of factoring expressions is a fundamental concept in algebra, enabling us to simplify complex expressions and solve equations more efficiently. In this article, we will delve into the specifics of factoring the expression 4x2 + 23x + 72, breaking down the steps involved in simplifying it.

Understanding the Basics of Factoring

Factoring involves expressing an algebraic expression as a product of simpler expressions, known as factors. This technique is essential in solving quadratic equations, finding the roots of polynomials, and simplifying complex expressions. To factor an expression, we need to identify the greatest common factor (GCF) of the terms and then group the terms accordingly.

Identifying the Greatest Common Factor (GCF)

The first step in factoring the expression 4x2 + 23x + 72 is to identify the greatest common factor (GCF) of the terms. The GCF is the largest expression that divides each term of the expression without leaving a remainder. In this case, the GCF is 1, since there is no common factor among the terms.

Factoring by Grouping

Since the expression 4x2 + 23x + 72 cannot be factored using the GCF, we will use the factoring by grouping method. This method involves grouping the terms into pairs and then factoring out the common factor from each pair.

Grouping the Terms

We will group the terms 4x2 and 23x, and then group the term 72.

(4x2 + 23x) + 72

Factoring Out the Common Factor

Now, we will factor out the common factor from each group.

(4x(x + 23/4)) + 72

However, this is not the factored form we are looking for. We need to further simplify the expression.

Factoring Quadratic Expressions

To factor the expression 4x2 + 23x + 72, we can use the quadratic formula or complete the square. However, in this case, we will use the factoring method for quadratic expressions.

Factoring the Quadratic Expression

We will look for two numbers whose product is 288 (4 × 72) and whose sum is 23. These numbers are 16 and 18, since 16 × 18 = 288 and 16 + 18 = 34 (which is not equal to 23). However, if we look at the product of the coefficients of x (4x2 and 23x), we can see that 4 × 18 = 72 and 4 + 16 = 20 (which is close to 23). Therefore, we can rewrite the expression as:

4x2 + 20x + 3x + 72

Factoring by Grouping Again

We will group the terms 4x2 and 20x, and then group the terms 3x and 72.

(4x2 + 20x) + (3x + 72)

Factoring Out the Common Factor Again

Now, we will factor out the common factor from each group.

4x(x + 5) + 3(x + 24)

Writing the Final Factored Form

Combining the two groups, we can write the final factored form of the expression as:

(4x + 3)(x + 24)

Therefore, the simplified expression of 4x2 + 23x + 72 is (4x + 3)(x + 24).

Conclusion and Next Steps

In this article, we have demonstrated the steps involved in factoring the expression 4x2 + 23x + 72. By understanding the basics of factoring, identifying the greatest common factor, and using factoring by grouping, we were able to simplify the expression into its factored form. We encourage you to practice factoring different expressions to become more proficient in this technique.

Now, we invite you to share your thoughts and ask questions in the comments section below. How do you approach factoring expressions? What techniques do you find most useful? Share your experiences and let's continue the conversation!