Combining and Simplifying 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.

Combining polynomials involves adding or subtracting them by combining like terms, which are terms that have the same variable raised to the same power.

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

To add polynomials, combine like terms by adding their coefficients. For example, $$(2x^2 + 3x) + (4x^2 + 5x) = (2+4)x^2 + (3+5)x = 6x^2 + 8x$$ .

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

The product of two binomials can be found using the distributive property (FOIL method). For example, $$(a + b)(c + d) = ac + ad + bc + bd$$ .

The square of a binomial is given by the formula $$(a + b)^2 = a^2 + 2ab + b^2$$ . For example, $$(x + 4)^2 = x^2 + 8x + 16$$ .

To multiply a monomial by a polynomial, distribute the monomial to each term in the polynomial. For example, $$3x(2x^2 + 4x) = 6x^3 + 12x^2$$ .

The result is $$x^3 - 3x^2 - 25x - 21$$ .

The result is $$6x^2 + 16x + 8$$ .