Polynomial and Rational Expressions: Cheat Sheet

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Essential Concepts

A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.
We can add and subtract polynomials by combining like terms.
To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products.
FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials.
A perfect square trinomial is a type of polynomial that results from squaring a binomial. It is called a “perfect square” because it is the exact square of a binomial expression.
When you multiply a binomial by another binomial that contains the same terms but with opposite signs, the result is known as the difference of squares.

The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem.
Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term.
Trinomials can be factored using a process called factoring by grouping.
Perfect square trinomials and the difference of squares are special products and can be factored using equations.
The sum of cubes and the difference of cubes can be factored using equations.
Polynomials containing fractional and negative exponents can be factored by pulling out a GCF.

Rational expressions are formed when one polynomial is divided by another, resulting in a fraction-like form where the numerator and the denominator are both polynomials.
Rational expressions can be simplified by cancelling common factors in the numerator and denominator.
We can multiply rational expressions by multiplying the numerators and multiplying the denominators.
To divide rational expressions, multiply by the reciprocal of the second expression.
Adding or subtracting rational expressions requires finding a common denominator.

perfect square trinomial	[latex]{\left(x+a\right)}^{2}=\left(x+a\right)\left(x+a\right)={x}^{2}+2ax+{a}^{2}[/latex]
difference of squares	[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]
perfect square trinomial	[latex]{a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}[/latex]
sum of cubes	[latex]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]
difference of cubes	[latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex]

coefficient: any real number [latex]{a}_{i}[/latex] in a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

difference of squares: the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign


factor by grouping: a method for factoring a trinomial of the form [latex]a{x}^{2}+bx+c[/latex] by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression

greatest common factor: the largest polynomial that divides evenly into each polynomial

leading coefficient
the coefficient of the leading term


least common denominator: the smallest multiple that two denominators have in common

monomial
a polynomial containing one term

perfect square trinomial: the trinomial that results when a binomial is squared



polynomial: a sum of terms each consisting of a variable raised to a nonnegative integer power

term of a polynomial: any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]