Operations with Polynomials

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.

Adding polynomials involves combining like terms, which are terms that have the same variable raised to the same power. For example, $$(2x^2 + 3x) + (4x^2 + 5) = (2x^2 + 4x^2) + 3x + 5 = 6x^2 + 3x + 5$$ .

Subtracting polynomials involves distributing the negative sign and then combining like terms. For example, $$(3x^2 + 2) - (x^2 + 4) = 3x^2 + 2 - x^2 - 4 = 2x^2 - 2$$ .

To multiply two polynomials, use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial, then combine like terms.

The result is $$x^2 + 5x + 6$$ .

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.

A monomial is a polynomial with only one term. For example, $$5x^2$$ and $$-3xy$$ are monomials.

A binomial is a polynomial with exactly two terms. For example, $$x + 5$$ and $$3x^2 - 2x$$ are binomials.

A trinomial is a polynomial with exactly three terms. For example, $$x^2 + 3x + 2$$ is a trinomial.

The formula is $$(a + b)(a - b) = a^2 - b^2$$ .