Category: Monomials and Polynomials

Monomials

Monomial is an algebraic expression, that can have the following 3 "parts":

• Number (it is called coefficient of monomial)
• Variables, raised to non-negative integer powers
• Operations of multiplication (they "separate" variables)

$$$\color{green}{\underbrace{15}_{\text{number (coefficient)}}}\color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{x^2}_{\text{variable}}} \color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{y^3}_{\text{variable}}}$$$

Binomials

Binomial is a sum/difference of TWO monomials.

Both monomials are called terms.

Examples of binomials are:

• $$${5}+{x}$$$
• $$${2}{{x}}^{{2}}+{{y}}^{{3}}{x}$$$
• $$${10}{x}{{y}}^{{3}}-{z}{{y}}^{{2}}{x}$$$

Examples of expressions, that are not binomials:

Polynomials

Polynomial is a monomial or sum/difference of monomials.

Monomials are called terms of the polynomial.

Note, that binomial is also polynomial.

Examples of polynomial are:

• $$${5}+{x}+{{x}}^{{2}}$$$
• $$${2}{{x}}^{{2}}+{{y}}^{{3}}{x}+{z}{{y}}^{{3}}$$$
• $$${10}{x}{{y}}^{{3}}-{z}{{y}}^{{2}}{x}$$$

Examples of expressions, that are not polynomials:

Adding Polynomials

To add polynomials we need to combine (add) like terms using distributive property of multiplication.

Recall, that like terms are terms that contain same variables, raised to the same powers. In other words, like terms are "like" each other. For example, $$${5}{{x}}^{{3}}{{y}}^{{2}}$$$ and $$${2}{{x}}^{{3}}{{y}}^{{2}}$$$.

Subtracting Polynomials

Subtraction of polynomials is quite similar to addition of polynomials. You just need to be careful with minus sign.

Similarly to adding, we can subtract polynomials either horizontally or vertically.

Multiplying Monomials

Monomials can be multiplied in the same manner as numbers.

To multiply monomials, we use commutative property of multiplication and properties of exponents.

Let's start from a simple example, involving only one variable.

Multiplying Polynomials by Monomial

To multiply polynomial by monomial, one should use distributive property of multiplication.

Then, just multiply monomials and you're done.

Example 1. Multiply $$${\left({2}{x}\right)}{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}$$$.

Multiplying Binomials (FOIL)

Binomials can be multiplied very easy by using FOIL.

FOIL method is derived by applying distributive property of multiplication three times.

Indeed, suppose we want to find $$${\left({a}+{b}\right)}{\left({c}+{d}\right)}$$$.

Square of Sum and Difference

Square of sum and difference:

$$$\color{purple}{\left(a\pm b\right)^2=a^2\pm 2ab+b^2}$$$

Let's see how to derive it.

Recall, that exponent is just repeating multiplication.

Thus, we can write that $$${{\left({a}+{b}\right)}}^{{2}}={\left({a}+{b}\right)}{\left({a}+{b}\right)}$$$.

Multiplying Polynomials

To multiply polynomials, you need to multiply each term of first polynomial by each term of another polynomial, then add resulting products, simplify and combine like terms (if possible).

As was stated in FOIL note, FOIL is not applicable for multiplying general polynomials (only binomials, which are polynomials with two terms).

Cube of Sum and Difference

Cube of sum and difference:

$$$\color{purple}{\left(a\pm b\right)^3=a^3\pm 3a^2 b+3ab^2\pm b^3}$$$

Let's see how to derive it.

Recall, that exponent is just repeating multiplication.

Thus, we can write that $$${{\left({a}+{b}\right)}}^{{3}}={\left({a}+{b}\right)}{{\left({a}+{b}\right)}}^{{2}}$$$.

Dividing Monomials

Monomials can be divided in the same manner as numbers.

To divide monomials, we use properties of fractions and properties of exponents.

Let's start from a simple example, involving only one variable.

Dividing Polynomials by Monomial

To multiply polynomial by monomial, one should use sum rule for fractions in reverse direction: $$$\frac{{{a}+{b}}}{{c}}=\frac{{a}}{{c}}+\frac{{b}}{{c}}$$$ (in fact, there can be more than two terms in the numerator).

Polynomial Long Division

Polynomial Long Division is a technique for dividing polynomial by another polynomial. It works in the same way as long division of numbers, but here you are dealing with variables.

You perform division step by step, by "guessing" terms of a quotient. Division is finished, when degree of the result is less than degree of the divisor.

Synthetic Division

Synthetic Division is a method, similar to polynomial long division, but it requires less writing and fewer calculations. However, it can be used only for dividing polynomial in one variable by linear polynomial $$${x}-{a}$$$.