For The Polynomial $6xy^2 - 5x^2y^7 + 9x^2$ To Be A Trinomial With A Degree Of 3 After It Has Been Fully Simplified, What Is The Missing Exponent Of $y$ In The Second Term?A. 0 B. 1 C. 2 D. 3
Introduction
Polynomials are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will delve into the world of polynomials and explore the process of simplifying them. We will also focus on a specific problem that requires us to find the missing exponent of y in the second term of a polynomial to make it a trinomial with a degree of 3.
What is a Polynomial?
A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x and y, and the coefficients are numbers that are multiplied by the variables.
Types of Polynomials
There are several types of polynomials, including:
- Monomials: A polynomial with only one term, such as 3x or 2y.
- Binomials: A polynomial with two terms, such as x + 3 or 2y - 4.
- Trinomials: A polynomial with three terms, such as x + 2y + 3 or 2x - 3y + 4.
Simplifying Polynomials
Simplifying polynomials involves combining like terms and eliminating any unnecessary terms. To simplify a polynomial, we need to follow these steps:
- Combine like terms: Combine any terms that have the same variable and exponent.
- Eliminate unnecessary terms: Remove any terms that have a coefficient of zero.
- Simplify the expression: Simplify the expression by combining any remaining like terms.
The Problem
The problem we are given is a polynomial with two terms: $6xy^2 - 5x2y7 + 9x^2$. We are asked to find the missing exponent of y in the second term to make it a trinomial with a degree of 3.
Step 1: Identify the Degree of the Polynomial
To solve this problem, we need to identify the degree of the polynomial. The degree of a polynomial is the highest exponent of any variable in the polynomial. In this case, the degree of the polynomial is 7, since the highest exponent of y is 7.
Step 2: Determine the Missing Exponent
To make the polynomial a trinomial with a degree of 3, we need to find the missing exponent of y in the second term. Since the degree of the polynomial is 7, we need to find an exponent of y that will reduce the degree of the polynomial to 3.
Step 3: Simplify the Polynomial
To simplify the polynomial, we need to combine like terms and eliminate any unnecessary terms. In this case, we can combine the terms with the same variable and exponent.
Step 4: Find the Missing Exponent
To find the missing exponent of y, we need to analyze the terms in the polynomial. The first term has an exponent of 2, and the third term has an exponent of 0. To make the polynomial a trinomial with a degree of 3, we need to find an exponent of y that will reduce the degree of the polynomial to 3.
The Solution
After analyzing the terms in the polynomial, we can see that the missing exponent of y is 3. This is because the exponent of y in the second term is 7, and we need to reduce the degree of the polynomial to 3. By setting the exponent of y in the second term to 3, we can make the polynomial a trinomial with a degree of 3.
Conclusion
In conclusion, simplifying polynomials is an essential skill for any math enthusiast. By following the steps outlined in this article, we can simplify polynomials and find the missing exponent of y in the second term to make it a trinomial with a degree of 3. The missing exponent of y is 3.
Answer
The missing exponent of y in the second term is 3.
Final Answer
Q: What is the difference between a monomial, binomial, and trinomial?
A: A monomial is a polynomial with only one term, such as 3x or 2y. A binomial is a polynomial with two terms, such as x + 3 or 2y - 4. A trinomial is a polynomial with three terms, such as x + 2y + 3 or 2x - 3y + 4.
Q: How do I simplify a polynomial?
A: To simplify a polynomial, you need to combine like terms and eliminate any unnecessary terms. This involves following these steps:
- Combine like terms: Combine any terms that have the same variable and exponent.
- Eliminate unnecessary terms: Remove any terms that have a coefficient of zero.
- Simplify the expression: Simplify the expression by combining any remaining like terms.
Q: What is the degree of a polynomial?
A: The degree of a polynomial is the highest exponent of any variable in the polynomial. For example, in the polynomial 3x^2 + 2y^3, the degree is 3 because the highest exponent of any variable is 3.
Q: How do I find the missing exponent of y in a polynomial?
A: To find the missing exponent of y in a polynomial, you need to analyze the terms in the polynomial and determine which exponent will reduce the degree of the polynomial to the desired value. In the problem we discussed earlier, we needed to find the missing exponent of y in the second term to make it a trinomial with a degree of 3.
Q: What is the difference between a polynomial and an expression?
A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An expression is a general term that can include variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division.
Q: Can I simplify a polynomial with variables and fractions?
A: Yes, you can simplify a polynomial with variables and fractions. To do this, you need to follow the same steps as before: combine like terms, eliminate unnecessary terms, and simplify the expression.
Q: How do I simplify a polynomial with negative exponents?
A: To simplify a polynomial with negative exponents, you need to rewrite the polynomial with positive exponents. This involves using the rule that a^(-n) = 1/a^n.
Q: Can I simplify a polynomial with imaginary numbers?
A: Yes, you can simplify a polynomial with imaginary numbers. To do this, you need to follow the same steps as before: combine like terms, eliminate unnecessary terms, and simplify the expression.
Q: How do I check my work when simplifying a polynomial?
A: To check your work when simplifying a polynomial, you need to plug your answer back into the original polynomial and verify that it is true. You can also use a calculator or computer program to check your work.
Q: What are some common mistakes to avoid when simplifying polynomials?
A: Some common mistakes to avoid when simplifying polynomials include:
- Forgetting to combine like terms: Make sure to combine any terms that have the same variable and exponent.
- Not eliminating unnecessary terms: Remove any terms that have a coefficient of zero.
- Not simplifying the expression: Simplify the expression by combining any remaining like terms.
- Not checking your work: Plug your answer back into the original polynomial and verify that it is true.
Conclusion
Simplifying polynomials is an essential skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify polynomials and find the missing exponent of y in the second term to make it a trinomial with a degree of 3.