Adding and Subtracting 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, $$(3x^2 + 2) - (x^2 + 4) = (3x^2 - x^2) + (2 - 4) = 2x^2 - 2$$ .

To combine like terms, identify terms with the same variable and exponent, then add or subtract their coefficients. For example, in $$3x^2 + 4x^2 - 2x + 5$$ , combine $$3x^2 + 4x^2 = 7x^2$$ , resulting in $$7x^2 - 2x + 5$$ .

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

A coefficient is a numerical factor in a term of a polynomial. For example, in the term $$5x^2$$ , the coefficient is 5.

A constant term is a term in a polynomial that does not contain any variables. For example, in the polynomial $$3x^2 + 4x + 5$$ , the constant term is 5.

First, distribute the negative sign: $$-7 - 8p - 4p^2 + 2 + 4p^4 - 4p^2$$ . Then combine like terms: $$4p^4 - 8p^2 - 8p - 5$$ .

First, distribute the negative sign: $$3a^2 - 4 - a^3 + 6a - 5 + 8a^3$$ . Then combine like terms: $$7a^3 + 3a^2 + 6a - 9$$ .

Combine like terms: $$(-2x^4 + 4x^4) + (-7x^3 + 4x^3) = 2x^4 - 3x^3$$ .