Polynomial Functions

• What are Polynomial functions?• Definition: Polynomial functions are defined and

continuous on all real numbers. • Example of a polynomial function: f(x)= 4x^3-5x-1/2 (f is

a polynomial function of degree 3 with leading coefficient 4)

• Topics we will be discussing: Quadratics, End behavior, Zeros, Rational root theorum, synthetic/ long division, factor theorum, remainder theorum, complex numbers, factoring over the reals, polynomial inequalities, rational zero theorum

• Quadratic function: f(x)=ax^2+bx+c• Definiton: a quadratic function is a polynomial function of degree 2• Graph: the graph of every quadratic function is an upward or

downward opening parabola. This is because the graph of any quadratic function can be obtained from the graph f(x)=x^2 by 2 sequence of translations, reflections, stretches, and shrinks.

• Example 1: describe how to transform the graph f(x)=x^2 into –(1/2)x^2+3. –(1/2)x^2+3 is obtained by vertically shrinking the graph f(x)=x^2 by ½, reflecting over the x-axis, then translating the graph 3 units up.

• Vertex form of a quadratic function: f(x)= a(x-h)^2+k. This vertex form for a quadratic function makes it easy to identify the vertex and axis of the graph of the function, and to sketch the graph. Using algebra to describe the graph of a quadratic function can be seen in example 2.

• Example 2:use completing the square to describe the graph of f(x)= 3x^2+12x+11.

• f(x)= 3x^2+12x+11, = 3(x^2+4x)+11, factor 3 from the x terms =3(x^2+4x+()-()))+11, prepare to complete the square

=3(x^2+4x+(2^2)-(2^2))+11, =3(x^2+4x+4)-3(4)+11, distribute the 3 =3(x+2)^2-1

The graph of f is an upward-opening parabola with vertex (-2,-1), axis of symmetry x=-2, x-intercepts x=-2 ± square root 3/3.

Original Graph Transformed Graph

• Leading Term Test for Polynomial End Behavior

• For any Polynomial function, the limits lim f(x) and lim f(x) are determined by the degree x ∞n of x - ∞ the polynomial and its leading coefficient.

an>0 n is odd, lim f(x)= -∞, lim f(x)= ∞

an<0 n is odd, lim f(x)= -∞, lim f(x)= -∞

X -∞ x ∞ X -∞ x ∞

an>0 n is even, lim f(x)=∞, lim f(x)= ∞

an<0 n is even, lim f(x)= -∞, lim f(x)= -∞

X -∞ x ∞ X -∞ x ∞

Rational Zero Theorem• Trickery: use the fraction. If you get the equation,

x^3+2x+1, the leading term integer goes on the bottom and the last integer goes on the top. Once you get the factors of the first and last integers, guess and test the factors using synthetic division to find the zeros. Be sure to do both the plus or minus of the factors.

• Example: x^4+4x^3-12x-9. take the integers 1 and -9. • + or - 1,3,9/+ or – 1• Guess and test synthetic division and you find that -1 and -

3 work. Then you take the remainder which is x^2-3• (x+1), (x+3), (x^2-3)• Zeros= -1, -3, + or – square root of 3

• Real world example: The number of video games that were sold by a company on their release date can be modeled by g(x)=2x^3+4x^2-2x, where g(x) is the number of games sold in hundreds and x is the number of hours after the first minute. How long did it take to sell 400 games?

• Factor out an x.• X(2x^2+4x-2)• + or – 1,2/+ or – 1,2• Use synthetic division to find zeros

Long Division

• Example 1:

• Example 2:This example has

a remainder of 9.

You then take the

remainder and put it

over the divisor.

Synthetic Division

• Example: 2x^3-3x^2-5x-12 divide by x=3

3] 2 -3 -5 -12

6 9 12

2 3 4 0

2x^2+3x+4• Trickery: MAMA ( Multiply Add Multiply Add)

Factor Theorem

• Theorem: A polynomial function f(x) has a factor x-k if and only if f(k)=0

• Example: Find the remainder when f(x)=3x^2+7x-20 is divided by x+4.

• f(-4)=3(-4)^2+7(-4)-20=48-28-20=0

Remainder Theorem

• Theorem: If a polynomial f(x) is divided by x-k, then the reminder is r=f(k).

• Example: Find the remainder when f(x)=3x^2+7x-20 is divided by x-2.

• f(2)=3(2)^2+7(2)-20=12+14-20=6

Zero’s

• Trickery: Zero’s are used quite often when dealing with polynomials. The best way to find zero’s is by factoring. Once an equation is broken down to the factored form of the polynomial, you set the equations equal to zero and find the value of the variable.

• Example: x^2-6x-27 Factor the polynomial down to (x-9)(x+3). Then set the two equations equal to zero (x-9=0)(x+3=0). That will give you two zero’s of x=9 and x=-3.

• Graphs: when incorporating zero’s with graphs, you must find the value of the zero. After that you plot the zero’s along the x-axis of the graph. It’s important that you watch for repeating zero’s when graphing though. If a zero has multiplicity it will change the path of the line that is being graphed. If the multiplicity(the exponent) is odd then the line passes through the x-axis. If the multiplicity is even it only touches the x-axis.

Zero’s

(x+2)^5

(x-3)^2

An example of multiplicity

Factoring over the Reals• Every polynomial function with real coefficients can be written as a

product of linear factors and irreducible quadratic factors, each with real coefficients.

• Example: f(x)= 3x^5-2x^4+6x^3-4x^2-24x+16• The Rational Zeros Theorem provides a lot of rational zeros that

might work. Using synthetic division, x=2/3 is a zero.

f(x)=(x-2/3)(3x^4+6x^2-24)=(x-2/3)(3)(x^4+2x^2-8)

=(3x-2)(x^2-2)(x^4)

=(3x-2)(x-square root of 2)(x+ square root of 2)(x^2+4)

Complex Numbers• A complex number is any number that can be written in the form a+bi.

Where a and b are real numbers. The real number a is the complex part, the real number b is the imaginary part, and a+bi is the standard form.

• Two complex numbers are equal if and only if their real and imaginary parts are equal; x+yi=2+5i, only if x=2 and y=5.

• Operations with complex numbers: if a+bi and c+di are two complex numbers then,

Sum: (a+bi)=(c+di)= (a+c) + (b+d)i

Difference: (a+bi)-(c+di)= (a-c)=(b-d)i• Example 1: • Sum=(7-3i)+(4-5i)=(7+4)+(-3+5)i=11+2i• Difference: (2-i)-(8+3i)=(2-8)+(-1-3)i=-6-4i• Example 2: multiplying (2+3i)(5-i)=2(5-i)+3i(5-i)

=10-2i+15i-3i^2

=10+13i-3(-1)

= 13+13i

Complex Numbers

• Example 3: Raising a complex number to a power.

• If z= ½ + square root 3/2i, find z^2 and z^3

z^2=(1/2 + square root 3/2i)(1/2 + square root 3/2i)

= ¼ + square root 3/4o + square root 3/4i + 3/4i^2

=1/4 + 2 square root 3/4i + ¾ (-1)

= -1/2 + square root 3/2i

Complex Numbers

• Z^3= z^2= z= (-1/2 + square root 3/2i)(1/2 + square root 3/2i)

= -1/4 – square root 3/4i + square root 3/4i + 3/4i^2

= -1/4 + 0i + ¾ (-i)

=-1

Polynomial Inequalities

• Every polynomial inequality can be written in the form f(x)>0, f(x)≥0, f(x)<0, f(x)≤0, where f(x) is a polynomial.

• Example 1: finding where a polynomial is zero, positive, or negative.• f(x)= (x+3)(x^2+1)(x-4)^2. Determine the real number

values of x that cause f(x) to be zero, positive, and negative. • Zero: the real zero’s of f(x) are -3(multiplicity 1) and 4( mulitiplicity

2). So f(x) is zero if x=-3 or x=4.• Positive: x^2+1 is positive for all real numbers. (x-4)^2 is positive

except x=4. x+3 is positive only if x>-3. So f(x) is positive if x>-3 and x≠4.

• Negative: f(x) is negative if x<-3.