4.3; All Zeros of Polynomials

• What is i

• Conjugate of a+bi is a-bi

• ALL numbers are considered complex numbers as all numbers can be written in a+bi form with b = 0.

Recall

• Use DeCarte's Rule of Signs to determine how many + or - real zeros AND imaginary zeros too.

• Imaginary zeros MUST come in pairs.

• Using Conjugate pairs & one given zero

• Determine the complex zeros (this means all of the them) of a polynomial function. RECALL----------------->

Today's Goals

All Zeros of a polynomial; this includes imaginaries.

Multiple Choice

Recall that $$i=\sqrt[]{-1}$$

Thus $$i^2=$$

• - 1

• 1

• meh

• Not this again

Multiple Choice

Determine the conjugate of 2-i

• 2 + i

• -2 + i

• - 2 - i

• 2 - i

Multiple Choice

Determine the conjugate of $$-3+4i$$

• -3-4i

• -3+4i

• 3-4i

• 3+4i

Now that we have....

...reviewed imaginary numbers with their conjugate form, lets look at how to use Decarte's beautiful Rule of Signs to determine the number of positive and negative real zeros we might have.

Then we can create a table of ALL types of zeros we might have.

Multiple Choice

Based on DeCartes' rule of signs how many positive and negative real zeros might f(x) have?

$$f\left(x\right)=x^2+3x-18$$

• 1 positive

  1 negative

• 0 positive

  2 negative

• 2 positive

  0 negative

• 0 positive

  0 negative

Multiple Choice

One zero is given determine the other zero.

• 1 +2i

• -1 +2i

• 1 - 2i

• - 1 - 2i

Multiple Choice

(a) how many zeros?

(b) how many positive and negative real zeros?

$$h\left(x\right)=x^3-3x^2+3x-9$$

• 3 total zeros

  3 positive

  0 negative

• 3 total zeros

  2 postive

  1 negative

• 3 total zeros

  1 positive

  2 negative

• 3 total zeros

  0 positive

  3 negative

Imaginary zeros MUST come in pairs so when we multiply them out we obtain a function with real numbers. Check out this video and the table --->

Be SURE to make a chart for the next few questions.

We can list the real zeros with the imaginary zeros.

Multiple Choice

If you have one imaginary zero you must have...?

• another; which is the conjugate of the original

• no other imaginary zeros

• you must have 3 more imaginary zeros

The next 4 slides are questions about the same polynomial function. Be sure to write it down with a chart. Keep in mind all zeros reduce by TWO

Dropdown

The amount of complex zeros (that means total zeros) is ​ ​

Select answer

Dropdown

The largest amount of positive real zeros is ​

Select answer

the largest amount of negative real zeros is​

Select answer

thus the amount of imaginary zeros is​

Select answer

Multiple Choice

If the largest amount of negative real zeros is 3 the next "reduced" of negative real zeors that is possible is which choice below?

• 2

• 1

• 4

• 0

Dropdown

If the amount of NEGATIVE real zeros is 1, then the amount of positive real zeros REMAINS ​

Select answer

and the amount of imaginary zeros must be​

Select answer

The next 3 slides are questions about the same polynomial function. Be sure to write it down with a chart. Keep in mind all zeros reduce by TWO

Dropdown

The total amount of complex zeros is ​ ​

Select answer

​

The largest number of positive real zeros is ​

Select answer

also ​

Select answer

is the number of negative real zeros with these two outcomes ​ there ​

Select answer

imaginary zeros

Dropdown

If there are 2 positive real zeros then there must still be ​

Select answer

negative real zero and ​

Select answer

imaginary zeros

Dropdown

Finally ​

Select answer

the least amount of positive real zeros,​

Select answer

MUST be the amount of negative real zeros and ​

Select answer

would be the number of imaginary zeros.

Try the next one

​

Multiple Choice

If $$x=3i$$ is a zero what MUST be the other zero?

• $$x=-3i$$

• $$x=3i$$

• $$x=1-3i$$

• $$x=1+3i$$

​Using complex zeros to find depressed equation

​Continued here...

NOW use long Division to obtain the "depressed" equation or other factor. First step is here. Finish this and go to the next two slides

Multiple Choice

Now we know that 3i and - 3i are complex zeros. What grouping below is a factor?

• $$x^2+9$$

• $$x^2-9$$

• $$x^2+3$$

• $$x=1+3i$$

Multiple Choice

After finishing long division using $$x^2+9\$$ we get the other factor of ....

• $$3x^2-5x-2$$

• $$3x^2+5x+2$$

• $$3x^2+5x-2$$

• $$-3x^2+5x-2$$

Multiple Choice

Finally the "depressed" equation of $$3x^2+5x-2$$ can be further factored to the following factors.

• $$\left(x-2\right)\left(3x-1\right)$$

• $$\left(x+2\right)\left(3x-1\right)$$

• $$\left(x-2\right)\left(3x+1\right)$$

Multiple Choice

of this function you can now list all of the zeros

• 3i, - 3i, -2, 1/3

• 3i, -3i, 2, 1/3

• 3i - 3i, -2, -1/3

Here comes a tougher problem. Write down the function for the next few slides

Multiple Choice

The listed comlex zero is - 1 +2i; what is another?

• -1 - 2i

• 1 - 2i

• - 1 +2i

• 1 +2i

Multiple Choice

Lets multiply our factors that are the imaginary zeros where

$$x=-1+2i\ \&\ x=-1-2i$$

With this as our zeros add 1 to both sides and then add/subtract the imaginary portion to get the following FACTORS

• $$\left(x+1-2i\right)\left(x+1+2i\right)$$

• $$\left(x-1-2i\right)\left(x-1+2i\right)$$

Multiple Choice

Now that we have multiplied these two imaginary zeros we should have a quadratic with all real zeros.

$$\left(x+1-2i\right)\left(x+1+2i\right)=?$$

• $$x^2+2x+5$$

• $$x^2+2x-3$$

Multiple Choice

Take your quadratic factor (divisor) and use long division to obtain the "depressed" equation or also called quotient. BUT first factor out that x so you are actually using

$$h\left(x\right)=2x^4+11x^3+27x^2+41x+15$$

as your dividend.

Chose the correct quotient below

• $$2x^2+7x+3$$

• $$2x^2-7x+3$$

• $$2x^2+7x-3$$

• $$x^2+7x-3$$

Multiple Select

Finally determine the last zeros by factoring.

$$2x^2+7x+3$$

If this is not factorable use the quadratic formula to find the real irrational zeros. Choose all that are the zeros from below

• $$x=-\frac{1}{2}$$

• $$x=3$$

• $$x=-3$$

• $$x=\frac{1}{2}$$

Multiple Choice

In summary all zeros are....

• $$\left\{-3,\ -\frac{1}{2},\ 0,\ -1+2i,\ -1-2i\right\}$$

• Pick the other one. ; )

1. Complex Zeros means ALL zeros; as 3 can be written as 3+0i

2. Imaginary zeros ALWAYS come in conjugate pairs

3. Decartes rule of signs is helpful to create a chart of the amount of real/imaginary zeros

4. When you can FACTOR

5. If you know one zero use division to find the depressed equation.

6. What else should you add to this summary?

​

​Time for summary...