The simplified sum of two polynomials can be a tricky concept to understand, particularly when the polynomials have multiple terms. This article provides a comprehensive explanation of the simplified sum of the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y. We will discuss what polynomials are, how to simplify the sum of two polynomials, and how to evaluate the simplified sum.

## What is a Polynomial?

A polynomial is an expression that consists of variables, coefficients, and exponents. Polynomials can have several terms, with each term having its own set of variables, coefficients, and exponents. The variables are the letters that represent the unknown values in the expression. The exponents are the powers to which the variables are raised. The coefficients are the numerical values that multiply the variables.

For example, the polynomial 2x2y3 + 4xy2 − 5y + 7 has four terms. The first term is 2x2y3, with variable x and y, coefficient 2, and exponents 2 and 3. The second term is 4xy2, with variable x and y, coefficient 4, and exponents 1 and 2. The third term is −5y, with variable y, coefficient −5, and exponent 1. The fourth term is 7, with coefficient 7 and no variables or exponents.

## Simplifying the Sum of Two Polynomials

The simplified sum of two polynomials is an expression in which the terms with the same variable and exponents are combined. To simplify the sum, we first group together all the terms with the same variable and exponents. Then, we add the coefficients of the grouped terms. If the sum of the coefficients is zero, the terms cancel each other out and the simplified sum does not include those terms.

To illustrate, let’s consider the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y. First, we group together all the terms with the same variable and exponents. The first polynomial has two terms with the same variable and exponents: 3x2y2 and −3x2y2. The second polynomial has two terms with the same variable and exponents: −2xy5 and 3x4y.

Next, we add the coefficients of the grouped terms. For the first polynomial, the sum of the coefficients is 3 + (−3) = 0. This means that the terms cancel each other out and the simplified sum of the first polynomial is 0. For the second polynomial, the sum of the coefficients is (−2) + 3 = 1. This means that the simplified sum of the second polynomial is 1x4y.

Therefore, the simplified sum of the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y is 1x4y.

## Evaluating the Simplified Sum

Once the simplified sum is determined, we can evaluate it by substituting numerical values for the variables. This will give us the numerical value of the simplified sum.

Let’s return to the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y. The simplified sum of these polynomials is 1x4y. If we substitute x = 2 and y = 3, the simplified sum becomes 1(2)4(3) = 24. This means that the numerical value of the simplified sum of the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y is 24.

## Conclusion

In conclusion, the simplified sum of the polynomials 3x2y2 − 2xy5 and −3x2y2 + 3x4y is 1x4y. This can be evaluated by substituting numerical values for the variables. For example, if we substitute x = 2 and y = 3, the simplified sum becomes 1(2)4(3) = 24.