Operations with polynomials Study Guide

Big Picture

Monomials and polynomials can contain numbers, variables, and exponents. They can be added, subtracted, multiplied, divided, and factored, just like real numbers. There are a few special products of polynomials that are important to know, such as the product of two binomials.

Key Terms

Monomial: A number, a variable with a positive integer exponent, or the product of a number and variable(s) with positive integer exponents.

Polynomial: A monomial or sum of monomials

Term: A part of the polynomial that is added or subtracted.

Coefficient: A number that appears in front of a variable.

Constant: A number without a variable.

Binomial: A polynomial with two terms.

Trinomial: A polynomial with three terms.

Standard Form: A form where the terms in the polynomial are arranged in order of decreasing power (exponents decrease from left to right)

Leading Coefficient: The coefficient of the term with the greatest power.

Degree of a Monomial: Sum of the exponents in the monomial.

Degree of a Polynomial: The greatest degree of the terms.

Like Terms: Terms in the polynomial with the same exponents (coefficients could be different).

Terminology

Examples of monomials:

$7, \frac{1}{2} x, 3 a^{2} b$

These are not monomials:

$\frac{3}{x}, 2^{a}, x^{- 1}$

A polynomial is made up of different terms that contain positive integer powers of the variables.

A term can be a coefficient with a variable or just a constant.
A polynomial with only two terms is called binomial, and a polynomial with only three terms is called a trinomial.
If the terms are written in standard form so that the exponents decreased from left to right, the first coefficient is the leading coefficient.
‍

$4$ is the coefficient of $x^{3}$ and is the leading coefficient
Degrees:
$4 x^{3}$ has degree $3$
$2 x^{2}$ has degree $2$
$- 3 x$ has degree $1$
$1$ has degree $0$
The degree of the polynomial is $3$

Addition & Subtraction of Polynomials

To add $2$ or more polynomials, write their sum and combine like terms. Once there are no more like terms, thepolynomial is simplified.
To subtract $1$ polynomial from the other, add the opposite of each term of the polynomial we are subtracting.

Example: $(4 x^{2} - 3 x y + 2) + (2 x^{3} + 5 y) - (x^{2} + 5 x y - 3)$

Group like terms: $(2 x 3) + (4 x^{2} - x^{2}) + (- 3 x y - 5 x y) + (5 y) + (2 - (- 3))$

Simplify: $2 x^{3} + 3 x^{2} - 8 x y + 5 y + 5$

Multiplication of Polynomials

Multiplying Monomials

Multiply the coefficients as we would any number and use the product rule for exponents.
The product rule for exponents is $x^{n} ∙ x^{m} = x^{n + m} .$

Multiplying Monomials

Use the distributive property so that every term in one polynomial is multiplied by every other term in the other polynomial.
The distributive property is $a (b + c) = a b + a c$ .

Another method is called FOIL. If given $(a + b) (c + d)$ :

Multiply the First terms in each polynomial $(a, c)$
Multiply the Outermost terms in each polynomial $(a, d)$
Multiply the Innermost terms in each polynomial $(b, c)$
Multiply the Last terms in each polynomial $(b, d)$
Combine any like terms

So $(a + b) (c + d) = a c + a d + b c + b d$

Polynomials can be multiplied vertically, similar to vertical multiplication with regular numbers.

Example: $(a + b) (c + d)$

$\begin{array}{r} a + b \\ \times & c + d \\ a d + b d \\ + a c + b c \\ a c + a d + b c + b d \end{array}$

Symbol

Meaning

Add

Substract

X or .

Multiply

\div

or /

Divide

\sqrt{}

\sqrt[n]{}

square root, nth root

| |

Absolute value

Equals

≠

Not equal

≈

Approximately equal

<, ≤
‍

Less than, less than or equal to

>, ≥
‍

Greater than, greater than or equal to

{ }

Set symbol

\in

An element of a set

( ), [ ]

Group symbols

Division of Polynomials

Dividing Monomials

Write as a fraction and use the quotient of powers.
The quotient rule for exponents is $\frac{x^{n}}{x^{m}} = x^{n - m}$

Dividing Polynomials

To divide a polynomial by a monomial, we can divide each term in the numerator by the monomial.
Example:
$\frac{3 x^{3} + 6 x - 1}{x} = \frac{3 x^{3}}{x} + \frac{6 x}{x} - \frac{1}{x} = 3 x^{2} + 6 - \frac{1}{x}$
To divide a polynomial by a binomial, use long division.
Dividend $\div$ Devisor = Quotient + $\frac{R e m a i n d e r}{D i v i s o r}$
The dividend is the numerator, and the divisor isthe denominator.

For example: $\frac{x^{2} + 4 x + 5}{x + 3}$

$\begin{aligned} x + 1 \\ x + 3 x^{2} + 4 x + 5 \\ \underset{―}{- x^{2} - 3 x} \\ x + 5 \\ \underset{―}{- x - 3} \\ 2 \end{aligned}$

So $\frac{x^{2} + 4 x + 5}{x + 3} = x + 1 + \frac{2}{x + 3}$

Tips:

Rewrite the polynomial in standard form.
Write any missing terms with zero coefficients.
Example: Rewrite $2 x^{2} + 3$ as $2 x^{2} + 0 x + 3$

Special Products of Polynomials

Square of a Binomial

$(a + b)^{2} = (a + b) (a + b) = a^{2} + a b + a b + b^{2} = a^{2} + 2 a b + b^{2}$

You can also remember the square of the binomial by drawing this diagram:

The area of the square is $(a + b) (a + b) = (a + b)^{2}$

The area can be found by adding up the four smaller squares and rectangles.
$(a + b) (a + b) = a^{2} + 2 a b + b^{2}$

We can also find $(a - b)^{2}$ by replacing $b$ with $- b$ :

$(a - b)^{2} = (a - b) (a - b) = a^{2} - a b - a b - b^{2} = a^{2} - 2 a b + b^{2}$

Sum and Difference Patterns

$(a + b) (a - b) = a^{2} + a b - a b + b^{2} = a^{2} - b^{2}$

$a$ and $b$ can represent numbers, variables, or variable expressions.

Algebra I

Operations with Polynomials

Big Picture

Key Terms

Terminology

Addition & Subtraction of Polynomials

Algebra I

Operations with Polynomials cont.

Multiplication of Polynomials

Multiplying Monomials

Multiplying Monomials

Division of Polynomials

Dividing Monomials

Dividing Polynomials

Special Products of Polynomials

Square of a Binomial

Sum and Difference Patterns