Polynomials are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and division. Adding and subtracting polynomials is a basic operation in algebra that involves combining like terms to simplify the expression.
To add or subtract polynomials, we need to look for like terms. Like terms are terms that have the same variables and exponents. For example, 2x and 5x are like terms, while 2x and 5y are not. To add or subtract polynomials, we combine the coefficients of the like terms.
Let's look at an example:
3x^2 + 4x - 2 + 2x^2 - 3x + 1
To add these polynomials, we first group the like terms:
(3x^2 + 2x^2) + (4x - 3x) + (-2 + 1)
We then combine the coefficients of the like terms:
5x^2 + 1x - 1
So the answer is 5x^2 + x - 1.
When subtracting polynomials, we follow the same process, but we also need to distribute the negative sign to each term in the second polynomial before combining the like terms.
Let's look at another example:
4x^3 - 6x^2 + 3x - 2 - (2x^3 + 5x^2 - 2x + 1)
To subtract these polynomials, we first distribute the negative sign:
4x^3 - 6x^2 + 3x - 2 - 2x^3 - 5x^2 + 2x - 1
We then group the like terms:
(4x^3 - 2x^3) + (-6x^2 - 5x^2) + (3x + 2x) + (-2 - 1)
And finally, we combine the coefficients of the like terms:
2x^3 - 11x^2 + 5x - 3
So the answer is 2x^3 - 11x^2 + 5x - 3.
In summary, adding and subtracting polynomials involves combining like terms by adding or subtracting their coefficients. We need to be careful to distribute the negative sign when subtracting polynomials.