Operations with Polynomials

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:

• $$7,\frac{1}{2}x, 3a^2b$$

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:
• $$4x^3$$ has degree $$3$$
• $$2x^2$$ has degree $$2$$
• $$-3x$$ has degree $$1$$
• $$1$$ has degree $$0$$
• The degree of the polynomial is $$3$$

• 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: $$(4x^2-3xy+2) + (2x^3+5y) - (x^2+5xy-3)$$

Group like terms: $$(2x3) + (4x^2-x^2) + (-3xy-5xy) + (5y) + (2-(-3))$$

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

Operations with Polynomials cont.

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 $$a(b+c) = ab+ac$$.

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 $$(a+b)(c+d) = ac + ad + bc + bd$$

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

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

$$\begin{array}{r} &a+b\\ \times\!\!\!\!\!\!&c+d\\ \hline &ad+bd\\ +ac+bc\!\!\!\!\!\!&\\ \hline &ac+ad+bc+bd\\ \end{array}$$

Symbol
Meaning
+
-
Substract
X or .
Multiply
$$\div$$ or /
Divide
$$\sqrt{}$$ or $$\sqrt[n]{}$$
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:
$$\frac{3x^3 + 6x - 1}{x} = \frac{3x^3}{x} + \frac{6x}{x} - \frac{1}{x} = 3x^2 + 6 - \frac{1}{x}$$
• To divide a polynomial by a binomial, use long division.
• Dividend $$\div$$ Devisor = Quotient + $$\frac{Remainder}{Divisor}$$
• The dividend is the numerator, and the divisor isthe denominator.

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

$$\require{enclose} \begin{array}{rll} x+1 && \\[-3pt] x+3 \enclose{longdiv}{x^2 + 4x +5}\kern-.2ex \\[-3pt] \underline{-x^2 - 3x\phantom{00}} && \\[-3pt] x + 5\phantom{0} && \\[-3pt] \underline{\phantom{0}-x - 3\phantom{0}} && \\[-3pt] \phantom{0}2 && \\[-3pt] \end{array}$$

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

Tips:

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

Special Products of Polynomials

Square of a Binomial

$$(a+b)^2 = (a+b)(a+b) = a^2+ab+ab+b^2 = a^2+2ab+b^2$$

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.
• $$(a+b)(a+b) = \textcolor{#90cde2}a^\textcolor{#90cde2}2+2\textcolor{#e2c802}a\textcolor{#e2c802}b+\textcolor{#df2e24}b^\textcolor{#df2e24}2$$

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

$$(a-b)^2 = (a-b)(a-b) = a^2-ab-ab-b^2 = a^2-2ab+b^2$$

Sum and Difference Patterns

$$(a+b)(a-b) = a^2+ab-ab+b^2 = a^2-b^2$$

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