DOL Adding and Subtracting Polynomials

A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. For example, $$2x^2 + 3x - 5$$ is a polynomial.

Adding polynomials involves combining like terms from each polynomial. 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, $$(3x^2 + 4) - (x^2 + 2) = (3x^2 - x^2) + (4 - 2) = 2x^2 + 2$$ .

Like terms are terms that have the same variable raised to the same power. For example, $$3x^2$$ and $$5x^2$$ are like terms, while $$3x^2$$ and $$4x$$ are not.

To combine like terms, add or subtract their coefficients while keeping the variable part the same. For example, $$2x + 3x = 5x$$ .

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

The leading coefficient is the coefficient of the term with the highest degree. For example, in $$5x^4 + 3x^2 - 2$$ , the leading coefficient is 5.

The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$3x^3 + 2x^2 - x + 5$$ .

First, distribute the negative sign: $$12p^2 + 8p - 6 - 9p^2 + 2p - 5$$ . Then combine like terms: $$(12p^2 - 9p^2) + (8p + 2p) + (-6 - 5) = 3p^2 + 10p - 11$$ .

First, distribute the negative sign: $$2x^2 + 6x + 5 - 6x^2 - 3x - 5$$ . Then combine like terms: $$(2x^2 - 6x^2) + (6x - 3x) + (5 - 5) = -4x^2 + 3x$$ .