What is the factored form of 4×2 23x 72 – In the realm of algebra, the factorization of polynomials plays a pivotal role. In this article, we delve into the fascinating world of factoring, embarking on a journey to uncover the secrets of the factored form of 4x^2 + 23x + 72. Brace yourself for an enlightening exploration of mathematical concepts and their practical applications.
The journey begins with an introduction to the concept of factoring polynomials, followed by a step-by-step guide to factoring the given expression using the factoring by grouping technique. We will then identify the two factors of the polynomial and discuss their significance.
Factoring 4x2 + 23x + 72
Factoring polynomials involves expressing a polynomial as a product of simpler polynomials called factors. Our goal is to factor the given expression 4x 2+ 23x + 72 into two factors.
Factoring Techniques, What is the factored form of 4×2 23x 72
We can use the factoring by grouping technique, which involves grouping terms with common factors and then factoring out those common factors.
For the expression 4x 2+ 23x + 72, we can group the first two terms and the last two terms:
4x 2+ 23x + 72 = (4x 2+ 23x) + 72
Now, we can factor out the common factor 4x from the first group:
(4x 2+ 23x) + 72 = 4x(x + 23/4) + 72
Next, we need to find two numbers that multiply to give 72 and add to 23/4. These numbers are 18 and 4.
Factors of the Polynomial
Therefore, we can rewrite the expression as:
4x(x + 23/4) + 72 = 4x(x + 18/4 + 4/4) + 72
Now, we can group the last two terms and factor out the common factor 4:
4x(x + 18/4 + 4/4) + 72 = 4x((x + 18/4) + 1) + 72
Finally, we can factor out the common factor (x + 18/4) + 1:
4x((x + 18/4) + 1) + 72 = (4x + 18) – (x + 4)
Expanded Form
To verify that this is the correct factorization, we can expand it:
(4x + 18) – (x + 4) = 4x 2+ 16x + 18x + 72
4x 2+ 16x + 18x + 72 = 4x 2+ 23x + 72
Therefore, the factored form of 4x 2+ 23x + 72 is (4x + 18) – (x + 4).
Applications of Factoring
Factoring polynomials has numerous applications in mathematics and science. It can be used to:
- Solve equations
- Simplify expressions
- Find the zeros of a polynomial
- Graph polynomials
- Solve real-world problems
Key Questions Answered: What Is The Factored Form Of 4×2 23x 72
What is the significance of factoring polynomials?
Factoring polynomials simplifies complex expressions, reveals their structure, and enables us to solve equations, analyze functions, and make predictions in various mathematical and scientific applications.
How can factoring be used in real-world problems?
Factoring has practical applications in fields such as engineering, physics, and economics. For instance, it can be used to design bridges, analyze financial data, and optimize resource allocation.
