Polynomial Functions

A polynomial in x is a sum of monomials* in x.

Example:   5x3 − 4x² + 7x − 8

*A monomial in x is a single term of the form  axn,  where a is a real number and n is a whole number.

The following are monomials in x:5x3 ,   −6.3x,    2

The degree of a term of a polynomial

The degree of a term is the sum of the exponents of all the variables in that term.

Example 1:

 The term  5x³  is of degree 3 in the variable x.

Example 2:

 This term  2xy²z³ is of degree 1 + 2 + 3 = 6 in the variables

x, y, and z. 

Here are all possible terms of the 4th Here are all possible terms of the 4th degree in the variables degree in the variables xx and and yy::

xx44,  ,  xx³³yy,  ,  xx²²yy²,  ²,  xyxy³,  ³,  yy44

Degree of a polynomial

The degree of a polynomial is the degree of the leading term.

The degree of this polynomial  

5x³ − 4x² + 7x − 8   is 3.

Here is a polynomial of the first degree:  x − 2.

1 is the highest exponent.

Which of the following is a polynomial?  If an expression is a polynomial, name its degree.

x3 − 2x² − 3x − 4 

Polynomial of the 3rd degree.

3y² + 2y + 1

Polynomial of the 2nd degree.

Is this a polynomial?  If the expression is a polynomial, name its degree.

x2 – 2x + 1 x

Not a polynomial, because 1  = x-1

x

which is not a whole number power.

x

1

x

1

x

1

Polynomials Functions

A polynomial function has the form

y = a polynomial

A polynomial function of the first degree, such as  

y = 2x + 1,  is called a linear function.

A polynomial function of the second degree, such as  

y = x² + 3x − 2,  is called a quadratic function.

Polynomials Functions

A polynomial function of the third degree, such as  

y = x3 + 3x2 − 2x + 5,  is called a cubic function.

A polynomial function of the fourth degree, such as  

y = x4,  is called a quartic function.

Zeros

A zero of a polynomial function is an ‘x’ value for which ‘y’ = 0 (the x-intercept).

At these ‘x’ values, its graph either cuts or touches the ‘x’ axis.

The maximum number of zeros of a polynomial is the same as its degree.

Zeros of polynomial functions

The zeros of this polynomial function are the x-intercepts of the graph. The zeros are the solutions of x when f(x) = 0.

What are the zeros?

-4, -1, 3

What are the zeros?

Graphs of Polynomial FunctionsIn general, polynomial function graphs consist

of a smooth line with a series of hills and valleys.  The hills and valleys are called turning points.

The maximum possible number of turning points is one less than the degree of the polynomial.

Cubic FunctionsIn general, cubic functions are shaped like a "sideways S".

The shapes of their graphs can be quite varied, but they have the basic property that the graph can cross a horizontal line in at most 3 points. It may cross in fewer points, or it may not cross at all.

In this graph, there are 2 turning points, and the graph crosses the x-axis in 3 places.

  Graph of f(x) = ax3 + bx2 + cx + d

a>0 a<0

Quartic Functions

There are 3 turning points in this graph, and the graph crosses the x-axis in 3 places. The graph of a quartic polynomial can cross the x-axis in at most 4 points. It may cross in fewer points, or it may not cross at all.

Graph of f(x) = ax4 + bx3 +cx2 + dx + e

a>0 a<0

Polynomials can be factored, so polynomial functions can be written in factored form. Yay!

This is the graph of the polynomial function

f(x) = x2 + 4x + 4

In factored form:

f(x) = (x + 2)2

f(x) = (x + 2 )2

What is the x-intercept of this graph?

(-2, 0)

What is the zero of this graph?

-2

What is the solution of this equation? 0= (x + 2)2

- 2 = x

Do you see how they are all related?

Effects of different factors   Single factors  -- graph cuts through the x-

axis.

Effects of different factors   Squared factors -- tangent to the x-axis at

the point (c,0)

Effects of different factors

Cubed factors --  flatten out around the point (c,0)

Multiplicity of zeros

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial.

f(x) = (x + 2)2

This zero (-2) has a multiplicity of 2

since it is tangent to the x-axis, so the

factor of (x + 2) would appear twice

when the polynomial is factored.

Write the factors of the polynomial function with the graph of

The graph crosses the x-axis at (-6, 0) and (7, 0), so the factors will be

(x – 6) and (x + 7).

Since the graph cuts the x-axis, then each factor has a "multiplicity“ of 1.

End Behavior of the Graph of a Polynomial Function

If the degree of a polynomial function is even, then the left end of the graph and the right end of the graph in the same direction.

left end right end

left end right end

End Behavior of the Graph of a Polynomial Function

If the degree of a polynomial function is odd, then the left end of the graph and the right end of the graph are in opposite directions.

right end

left end

If the degree of the polynomial is odd AND the leading coefficient is

positive, the graph falls to the left and rises to the right:

If the degree is odd AND the leading coefficient is negative, the graph rises to the left and falls to the right.

If the degree is even AND the leading coefficient is

positive, the graph rises to the left and to the right.

If the degree is even AND the leading coefficient, is

negative, the graph falls to the left and to the right.

Which of the following could be the graph of a polynomial whose leading term is "–3x4"?

B

f(x) = x4 – x3 - 7x2 + x + 6 orf(x) = (x – 3)(x – 1)(x + 1)(x + 2)

Describe the graph of the above polynomial function.What is the end behavior of the graph? Why?Rises to the right and rises to the left because leading coefficient is positive

and degree is even.

Where does the graph intersect the x-axis and describe how the graph intersects the x-axis?

The graph intersects the x-axis at

(3, 0), (1, 0), (-1, 0) and (-2, 0).

Since the multiplicity of each zero is 1, then the graph cuts the x-axis at each x-intercept.

f(x) = x4 – x3 - 7x2 + x + 6 orf(x) = (x – 3)(x – 1)(x + 1)(x + 2)

Make a sketch of the graph.

f(x) = x3 – 5x2 – x + 5or f(x) = (x – 1)(x + 1)(x – 5)

Describe the graph of the above polynomial function.

What is the end behavior of the graph? Why?Rises to the right and falls to the left because leading coefficient is

positive and degree is odd.

Where does the graph intersect the x-axis and describe how the graph intersects the x-axis?

The graph intersects the x-axis at

(1, 0), (-1, 0) and (5, 0).

Since the multiplicity of each zero is 1, then the graph cuts the x-axis at each x-intercept.

f(x) = x3 – 5x2 – x + 5or f(x) = (x – 1)(x + 1)(x – 5)

Make a sketch of the graph of the above polynomial function.

f(x) = x3 – 2x2 - 15x + 36or f(x) = (x – 3)2 (x + 4)

Describe the graph of the above polynomial function.

What is the end behavior of the graph? Why?Rises to the right and falls to the left because leading coefficient is positive and

degree is odd.

Where does the graph intersect the x-axis and describe how the graph intersects the x-axis?

The graph intersects the x-axis at

(3, 0), (-4, 0).

Since the multiplicity of (x + 4) is 1 , then the graph cuts the x-axis at (-4, 0).

Since the multiplicity of (x – 3) is 2, then the graph is tangent to the x-axis at (3, 0).

f(x) = x3 – 2x2 - 15x + 36or f(x) = (x – 3)2 (x + 4)

Describe the graph of the above polynomial function.

f(x) = x3 - 5x2 + 7x - 3 orf(x) = (x – 3)(x – 1)2

Describe the graph of the above polynomial function.

f(x) = x4 + x3 - 2x2

orf(x) = x2(x + 2)(x – 1)

f(x) = -x3 – 9x2 – 27x – 27 orf(x) = -(x + 3)3

f(x) = -4x3 – 4x2 + 4x + 4 orf(x) = -4(x – 1)(x + 1)2

Write a possible equation for this polynomial function. The zeros are -6 and 7.

f(x) = (x + 6)2(x + 7)

The End!!!