PAM - Polynomials: Performing Arithmetic Operations and Graphing Lesson

Polynomials: Performing Arithmetic Operations and Graphing

What are Polynomials?

Polynomials are algebraic expressions with one or more unlike terms. Linear equations, the quadratic equations, exponential equations, and equations that have even higher exponents on the terms may all be written as polynomial equations if each term is written in polynomial form.

What may a polynomial term contain?

a. The terms may have different variables, coefficients of variables and constants, for example, terms could be x, 3x², 5xy, 9, ½, .6 or equation image indicator or other letters and numbers. Note that constants that are fractions, decimals, or roots are allowed.

b. Exponents may not be negative, so terms may not be of the form x^-3.

c. Terms may not contain radical expressions containing a variable, for example, equation image indicator = x^1/2, using the equivalent form of x^1/2 would not be acceptable as a polynomial term.

d. Terms may not contain rational expressions containing a variable in the denominator. For example, the term equation image indicator is not a valid polynomial term.

Remember that equivalent forms of writing fractional parts allow both sides of the equations to be multiplied by the denominator in order to create equations without denominators.

Polynomials are named by the polynomial term with the highest exponent. It is customary to always place the terms in descending order. Sometimes questions mix up the order, so put the polynomial in descending order of exponents prior to deciding on the degree.

Degree - the highest exponent found in the polynomial terms. All terms from the highest to the lowest (zero) on the constant may be in the polynomial or only some.

Leading Coefficient - the number in front of the term with the highest degree.

Let's look at an example of the above two words:

y = 3x² - 5x - 4

A polynomial of degree 2 and a leading coefficient of 3. A second order polynomial.

Polynomial Example	Degree	Type of Equation	Order	Alternate Name
y = 5x - 3.2	1	Linear	1^st Order	Line
y = x² - 4	2	Quadratic	2^nd Order	Parabola
y = x³ - 2x + 1	3	Cubic	3^rd Order
y = x⁴ + 5x²	4	Quartic	4^th Order
y = x⁵ + x³ -x	5	Quintic	5^th Order
. . .	. . .		. . .
y = xⁿ - x^n-1 + 7x² - x + 3	n		nth Order

There are more names than listed above, but not many are used. After the 3rd order polynomial which is called a cubic, most are just referred to by their degree or as the appropriate order.

The number of possible real number solutions for the polynomial equation is at most the degree of the equation. Remember, solutions are the zeros of the equation, the points where the equation crosses the x-axis.

Add Polynomials

Let's try a few quickly to make sure you understand the terminology and how to appropriately add polynomials. Remember adding a negative is subtracting.

Example 1: Given g(x) = -4x + 5 and h(x) = 6x - 2, find g(x) + h(x).

(-4x + 5) + (6x - 2) ( ) used to show the expressions g(x) and h(x)

-4x + 5 + 6x - 2 Nothing is multiplied as a coefficient to either ( ) so the ( ) are not needed.

2x + 3 Combine like terms.

g(x) + h(x) = 2x + 3 a completely new equation that we could call f(x).

The new equation may be given a function name of its own as the combination of two or more polynomial equations create a completely new polynomial equation.

In the example below, watch carefully the multiplication of the polynomial by a constant.

Example 2: Given f(x) = 3x² - 2x + 5 and g(x) = -4x + 5, find h(x) = f(x) - 3g(x).

(3x² - 2x + 5) - 3(-4x + 5) ( ) used to show the expressions f(x) and g(x)

(3x² - 2x + 5) + −3(-4x + 5) Alternate writing of the above letting the negative go with the coefficient 3.

3x² - 2x + 5 + 12x - 15 Distribute the -3.

3x² + 10x - 10 Simplify by combining like terms. The appropriate order of terms is always in descending exponent order.

f(x) - 3g(x) = 3x² + 10x - 10 a completely new equation h(x).

Example 3: Given the f(x) = 2x - 3xy + y² and g(x) = x³ + x - 4xy + 3xy², find f(x) + g(x) = h(x).

Sometimes it is easier to line up like terms under each other, then add.

Notice when I evaluate the equation this way, creating columns of like terms, it is easy to see what matches. All variables and their appropriate exponents must match in order to add terms together. The coefficients simply indicate how many of this type of term.

When we add them, we find the exponents on variables that match. Noticing in Example 3, y² could not be added to 3xy² because the variable x in 3xy² makes the terms not match. Likewise, the x³ and 3x terms cannot be added as the exponents on the x variables do not match.

Let's look at a simple example and examine the addition concept graphically.

Example 4: Find h(x) = f(x) + g(x).

ƒ(x) = x + 5 blue line

g(x) = x - 2 green line

h(x) = ƒ(x) + g(x) = 2x + 3 red line

Example 4: Find h(x) = f(x) + g(x).

ƒ(x) = x + 5 blue line

g(x) = x - 2 green line

h(x) = ƒ(x) + g(x) = 2x + 3 red line

Using our learned analysis skills, we can visually see that y-values for the f(x) and g(x) graphs at the same x-value sum to the y-value for the new equation; h(x) = f(x) + g(x). An easy x-value to check are x = 0. Check x = 2 on your own. The sum here should be 7.

At x = 0, f(x) = 0 + 5 = 5 creating the point (0, 5)

At x = 0, g(x) = 0 - 2 = -2 creating the point (0, -2)

At x = 0, h(x) = 2(0) + 3 creating the point (0, 3).

Adding the y-values for f(x) and g(x) yields

f(x) + g(x) = 5 + -2 = 3 = h(x)

Rules of Exponents

Let's stop and have a quick review for the rules of exponents prior to looking at multiplication or division of polynomials.

Rule 1: To add terms of a polynomial, the exponents and variables must match.

3xy² + 5xy² = 8xy² y² - 6y² = -5y²

Rule 2: When polynomial terms are multiplied, exponents are added and the same variables do not have to be in each term. A variable without an exponent shown has an understood exponent of 1.

Rule 3: Any item, a constant or a variable to the zero power (exponent) is 1.

Rule 4: A negative exponent on a variable or constant means that the variable or constant belongs in the opposite location, numerator or denominator.

Multiply Polynomials

To multiply polynomials simply use each term from the first polynomial times all terms in the second polynomial.

Example 5: Let f(x) = 2x + 4 and g(x) = 3x - 5. Find f(x) * g(x).

(2x + 4)(3x - 5)

Write each polynomial using ( ) to group and indicate that the complete polynomial is to be multiplied by the other complete polynomial.

2x(3x - 5) + 4(3x - 5)

Each term of the first polynomial is to be multiplied by each term of the second polynomial. Note that the exponent of 1 is on all x terms.

Distribute each term.

Simplify by combining like terms.

Using this technique of allows us to multiply different size polynomials easily.

View the video below to learn more on multiplying polynomials.

Composition of Polynomials

Now let's try a different thought process, called composition. We have inadvertently been doing this from the time of using function notation. Composition simply puts one item inside another to create a new entity, a new composition.

A real-world example is chocolate milk. You start with white milk in a glass and chocolate syrup, each separate parts on their own. When you put chocolate syrup in the glass you have put the chocolate syrup within the milk and when you combine them by stirring you have created chocolate milk, a new creation made up of two separate items.

We do the same thing in math with equations. We substitute for the x value numbers and we create a relationship, a point.

f(x) = x + 2

f(1) = 1 + 2 = 3, we have created the point on the graph (1, 3) by putting the number 1 in for the x term in the equation. A new composition.

We can create many more points by using different numbers for x.

Suppose we want to compose new items like the chocolate milk by putting one expression inside another expression.

Example 6: Suppose f(x) = x + 3 and g(x) = x². Find f(g(x)) = f º g(x).

This terminology is read f of g of x meaning to substitute for x whatever g(x) is equal to.

f(g(x)) = f(x²) = x² + 3 x² is substituted for x everywhere it is found in the f(x) = x + 3 equation.

Reasoning points:

a. The new equation we could call h(x) as the new entity, h(x) = x² + 3.

b. The new composition we quickly see is a parabola that goes upward and is shifted from 0 vertically up 3 units.

c. In comparison, the new h(x) equation changed a line with a y-intercept of 3 (original f(x)) to a parabola with a y-intercept of 3.

The video below will provide more examples of composition and simplifying a composed equation.

Factor Polynomials

Factoring polynomials allows us to be able to divide polynomials as well as find the zeros of the polynomials. We have done some basic factoring to find the zeros (the points where graphs cross the x-axis). Let's review the factoring rules that will be important.

Rule 1: The difference of squares. If the first term of a polynomial is a square and the second (last term) of the polynomial is a square, then quick factoring can occur.

x² - 9 25y⁴ - 16x²

(x - 3)(x + 3) (5y² - 4x)(5y² + 4x)

Note that the difference of squares takes the square root of each term and creates a factor with a minus and another with a plus. The beginning term is the same for each factor and the ending term is the same.

Try multiplying them back together. You will notice that the middle term drops out using the minus / plus combination in the factors.

Rule 2: Basic factoring of a trinomial (a three-term polynomial) requires

the first term in each factor multiplies to be the first term in the original equation
the last term in each factor multiplies to be the last term in the original equation
the middle term is the sum of the factors used in to create the last term of the original equation.

x² - 4x - 32 Use factors of 32

(x - 8)(x + 4) The sign of the larger number is the sign of the middle term. This is an equivalent form of the original equation.

Rule 3: Remember, if you cannot see quickly how to factor a trinomial, the quadratic formula will give you the solution and then you can create the factors.

x = 4 and x = -3

factors x - 4 = 0 x + 3 = 0

factored equation is (x - 4)(x + 3) = 0.

The video below will discuss Rule 2 and 3 above in more detail.

Divide Polynomials

Polynomial division requires either factoring, long division, or synthetic division to solve the division.

Division by factoring requires the numerator and/or the denominator to factor, to be written in equivalent form. In factoring, one of the factors would be in both equations, thus allowing the factors to become a form of 1.

Example 7: Simplify

$LaTeX: \frac{\left(x-2\right)\left(x-4\right)}{x-4}$ $\frac{\left(x-2\right)\left(x-4\right)}{x-4}$ Factored form of the numerator.

x - 2 Simplified answer. (x - 4)/(x - 4) = 1

On a graph, the domain of this function would be all real numbers, x ≠ 4, since 4 would make the original equation zero in the denominator. Division by 0 is illegal. The point (4, 2) would be shown on the graph as a circle indicating that the point does not exist (4 - 2 = 2)

This video will show how to divide polynomials using long division. Long division may have a remainder as well, providing a discussion of fractional parts with polynomial division.

The video below shows a quick division method as opposed to long division. Synthetic division will provide the same answer as long division using the coefficients of the polynomials only with multiplication and addition. The denominator must have only two terms.

Polynomial Expression Practice

Create Equivalent Rational Expressions

Now that we have reviewed rational numbers and polynomial division, let's look at equivalent rational expressions. Remember a rational expression is one that has a variable in the denominator.

How would we find the letter W to create an equivalent expression to The equivalent expression is W + Remember the minus means plus a negative number. Try solving for "W" yourself and then view the video to find out what "W" is.

Note that one side of the equation is in improper form (a fraction with the numerator larger than the denominator) and the other side is in proper form, number and then the fractional remainder. How do we know the second portion on the left side is the fractional remainder? The number is divided by the same denominator as the other side.

Graphing

It is important to examine a few more graphs to see patterns and understand the changes in graphs based on the higher order of polynomials. You will find that the rules, the reasoning foundations that you have learned continue to hold across other graphical representations.

Extending Knowledge Practice

View the video below to continue extending your reasoning and analysis skills, drawing conclusions from our existing knowledge.

In this next video we will use our knowledge and reasoning to exam how to sketch graphs given the x-intercepts.

Let's extend our graph analysis one more time by adding an intersection of a radical equation with a linear equation. Here we will find two solutions, but only one is a workable solution. A slightly different scenario than Example 8 in Rational and Radical Equations. Watch for the extraneous solution and how to weed it out of your final solution.

Example 8: Solve

Note that the complete expression x + 4 is under the square root symbol and the other side, the y-side, has another expression x - 2.

Square both sides. Note that you must square the complete expression x - 2, not each part separately.

x + 4 = x² - 4x + 4 (x - 2)(x - 2)

0 = x² - 5x Simplify by subtracting x and subtracting 4 from both sides.

0 = x(x - 5) Factor by finding the greatest common factor which is x.

x = 0 and/or x - 5 = 0 Set both factors = 0 as one or both of them must be zero for the factors to multiply to zero.

x = 0 and/or x = 5 The zeros or roots of the equation

Check your work:

Substitute in one of the zeros, so x = 0.

Red X

No, this is not correct as to be a function, the square root of 4 is 2, equation image indicator . See the graph below.

Substitute in the other zero, x = 5.

Correct. This one works and is the only solution as shown on the graph below.

The intersection of $LaTeX: y=\sqrt{x+4}$ $y=\sqrt{x+4}$ and LaTeX: y=x-2 $y=x-2$ is shown in the graph below.

Notes:

Each side of the equation was set to y = in order to find an intersection.
The only intersection of the graph of the two functions is at (5, 3).
The domain and range of domain x ≥ −4 and range y ≥ 0.
The domain and range of y = x - 2: a linear function, domain and range are all real numbers.

image graph
Each side of the equation was set to y = in order to find an intersection.

The only intersection of the graph of the two functions is at (5, 3).

The domain and range of domain x ≥ −4 and range y ≥ 0.

The domain and range of y = x - 2: a linear function, domain and range are all real numbers.

Important: Always check your answers for extraneous ones when dealing with even roots or exponents!

Review

Some forms to use to create equations based on the data given.

Linear

y-intercept form: y=mx+b

point-slope form:

Quadratic

general form: ax²+bx+c=0

vertex form: y=a(x-h)²+k

factored form: a(x-p)(x-q)=0

where p and q could be equal

Exponential

exponential form:

Square Root

looks like quadratic equation, but has a fractional exponent and lays on its side (remember this is one-sided to be a function)

Remember, these forms hold the keys to the basic understanding of all equations and how they work, regardless of the name an equation is given. Common understandings between these equations and graphs continue across all equations and graphs that you will encounter.

Take the leap and trust your knowledge and reasoning power to draw conclusions and infer the path to take to solve problems.

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