Simplifying Polynomial Expressions
In mathematics, polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and nonnegative integer exponents. Simplifying polynomials involves combining like terms to express the polynomial in its most compact form.
Let's consider the following expression:
(x^6 + 10x^5  6x^2  5x) + (9x^5 + 5x^3 + 9x^2 + x + 10)
To simplify this expression, we will follow these steps:

Identify like terms: Like terms are terms that have the same variable and exponent. In our expression, we have:
 x^6: This term is only present once.
 x^5: We have 10x^5 and 9x^5.
 x^3: We have 5x^3.
 x^2: We have 6x^2 and 9x^2.
 x: We have 5x and x.
 Constant: We have 10.

Combine like terms: We add or subtract the coefficients of the like terms.
 x^6: Remains as x^6.
 x^5: 10x^5  9x^5 = x^5
 x^3: Remains as 5x^3.
 x^2: 6x^2 + 9x^2 = 3x^2.
 x: 5x + x = 4x.
 Constant: Remains as 10.

Write the simplified expression: Putting it all together, we get the simplified form:
x^6 + x^5 + 5x^3 + 3x^2  4x + 10
Therefore, the simplified form of the given expression is x^6 + x^5 + 5x^3 + 3x^2  4x + 10.