Polynomial Algebra Review

Equations are made up of monomials and polynomials.

A monomial is a single variable or product of constants and variables, such as x, 2x, or  .

There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients. Polynomials are algebraic expressions which use addition and subtraction to combine two or more monomials.

Two terms make a binomial, three terms make a trinomial, and so on.

The degree of a monomial is the sum of the exponents of the variables.

The degree of a polynomial is the highest degree of any individual term.

To multiply two binomials, follow the FOIL method.

FOIL stands for: First: Multiply the first term of each binomial Outer: Multiply the outer terms of each binomial Inner: Multiply the inner terms of each binomial Last: Multiply the last term of each binomial Using FOIL,
.

To divide polynomials, begin by arranging the terms of each polynomial in order of one variable. You may arrange in ascending or descending order, but be consistent with both polynomials.

To get the first term of the quotient, divide the first term of the dividend by the first term of the divisor. Multiply the first term of the quotient by the entire divisor and subtract that product from the dividend. Repeat for the second and successive terms until you either get a remainder of zero or a remainder whose degree is less than the degree of the divisor.

If the quotient has a remainder, write the answer as a mixed expression in the form:
 .

Rational expressions are fractions with polynomials in both the numerator and the denominator; the value of the polynomial in the denominator cannot be equal to zero.

To add or subtract rational expressions, first find the common denominator, then rewrite each fraction as an equivalent fraction with the common denominator. Finally, add or subtract the numerators to get the numerator of the answer, and keep the common denominator as the denominator of the answer.

When multiplying rational expressions factor each polynomial and cancel like factors (a factor which appears in both the numerator and the denominator). Then, multiply all remaining factors in the numerator to get the numerator of the product, and multiply the remaining factors in the denominator to get the denominator of the product.

Remember – cancel entire factors, not individual terms.

To divide rational expressions, take the reciprocal of the divisor (the rational expression you are dividing by) and multiply by the dividend.

Rational Polynomial Functions

Below are patterns of some special products to remember: perfect trinomial squares, the difference between two squares, the sum and difference of two cubes, and perfect cubes.

Perfect Trinomial Squares:
 or
·

Difference Between Two Squares:
· 

Sum of Two Cubes:
 

Note: the second factor is NOT the same as a perfect trinomial square, so do not try to factor it further.

Between Two Cubes:
Again, the second factor is NOT the same as a perfect trinomial square.

Perfect Cubes:
 and 

In order to factor a polynomial, first check for a common monomial factor.

When the greatest common monomial factor has been factored out, look for patterns of special products: differences of two squares, the sum or difference of two cubes for binomial factors, or perfect trinomial squares for trinomial factors.

If the factor is a trinomial but not a perfect trinomial square, look for a factorable form, such as  or   . For factors with four terms, look for groups to factor.

Once you have found the factors, write the original polynomial as the product of all the factors.

Make sure all of the polynomial factors are prime. Monomial factors may be prime or composite.

Check your work by multiplying the factors to make sure you get the original polynomial.