Polynomial Operations

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$2x^2 + 3x - 5$$ is a polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$4x^3 + 2x^2 - x + 7$$ , the degree is 3.

The sum of two polynomials is obtained by adding their corresponding coefficients. For example, the sum of $$(2x^2 + 3x) + (4x^2 + 5)$$ is $$6x^2 + 3x + 5$$ .

To subtract polynomials, you subtract the coefficients of like terms. For example, $$(3x^2 + 4x) - (2x^2 + 5) = (3x^2 - 2x^2) + (4x - 0) - 5 = x^2 + 4x - 5$$ .

Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree, similar to numerical long division.

The result is $$3x - 4$$ .

Expanding a polynomial means to multiply out the factors and simplify the expression. For example, expanding $$3(2a + 5) + 5(a - 2)$$ results in $$11a + 5$$ .

Like terms are terms that have the same variable raised to the same power. For example, $$3x^2$$ and $$5x^2$$ are like terms.

Combining like terms involves adding or subtracting the coefficients of like terms to simplify the polynomial. For example, $$2x + 3x = 5x$$ .

A monomial is a polynomial with only one term, while a polynomial can have multiple terms. For example, $$3x$$ is a monomial, and $$3x + 2$$ is a polynomial.