What is the factored form of the polynomial 27x2y-43xy2 – Embark on a mathematical journey to uncover the secrets of factoring polynomials. Beginning with what is the factored form of the polynomial 27x2y – 43xy2, we will delve into the intricacies of this algebraic operation, exploring the steps, methods, and formulas that lead to its successful completion.
Through a comprehensive examination of common factors, grouping techniques, and verification methods, we will unravel the mysteries of polynomial factorization, empowering you with the knowledge to tackle even the most complex expressions with confidence.
Factoring the Polynomial: What Is The Factored Form Of The Polynomial 27x2y-43xy2
To factor the polynomial 27x 2y – 43xy 2, we need to find common factors between the terms and group them accordingly. Once grouped, we can factor out the common factors to obtain the factored form.
Common Factors
The first step is to identify the common factors between the terms 27x 2y and -43xy 2. Both terms have a common factor of xy, so we can write the polynomial as:
- x2y
- 43xy 2= xy(27x
- 43y)
Grouping Terms
Next, we need to group the terms in the polynomial so that each group has a common factor. We can group the terms as follows:
(27x2y
9xy2) + (-34xy 2)
Factoring by Grouping, What is the factored form of the polynomial 27x2y-43xy2
Now, we can factor out the common factor from each group:
xy(27x
- 9y)
- 34xy2
We can further factor the first group by extracting a common factor of 3:
xy(3(9x
- 3y))
- 34xy2
Finally, we can combine the factors to obtain the factored form of the polynomial:
xy(3(9x
- 3y)
- 34y)
Verifying the Factored Form
To verify if the obtained factored form is correct, we can expand it and compare it to the original polynomial:
xy(3(9x
- 3y)
- 34y)
= xy(27x
- 9y
- 34y)
= xy(27x
43y)
= 27x 2y
43xy2
Since the expanded form matches the original polynomial, the factored form is correct.
Question & Answer Hub
What is the significance of finding the factored form of a polynomial?
Factoring a polynomial simplifies its expression, making it easier to analyze, solve equations, and perform other mathematical operations.
How do I identify common factors in a polynomial?
Examine the coefficients and variables of each term to find the greatest common factor (GCF) that can be divided evenly into all terms.
What is the purpose of grouping terms in polynomial factorization?
Grouping terms with common factors allows you to factor out those factors and simplify the expression.
