Your Algebra Homework Can Now Be Easier Than Ever!

Real Zeros of Polynomial Functions

Review:

We started with quadratic functions (2nd degree polynomials)

Then we talked about polynomial functions . Defined roots /zeros.

If r is a real zero , then what do we know is a factor of the polynomial .

How does synthetic division relate to this?
------------------------------------------------
Now we will start putting all of these ideas together to find real zeros of a
polynomial
function, not given to us in factored form.

Factor Theorem: Let f be a polynomial function. Then (x-c) is a factor of f(x) if
and only if f(c) = 0.

De termine whether x-c is a factor of f. If it is, write f in factored form.

f (x) = 3x6 + 2x3 −176 ; c = -2

Now starting with a polynomial. How can we get the first zero?

Hopefully there is a zero that is not only real, but also rational.

Rational Zeros Theorem: Suppose we have a polynomial function and each
coefficient is an integer . Then if there is a rational zero, p/q, then p is a factor of
the last coefficient and q must be a factor of the first coefficient.

ex.f (x) = 2x3 +11x2 − 7s −6 So we can list all of the possible rational zeros
of f(x). They are all the ( positive and negative ) factors of 6, divided by all of the
factors of 2. Now we can look at the graph to see which one may work.

Once this is done we can use synthetic division to factor the polynomial. Why?

Let us review all of these steps .

We are given a polynomial function and want to find the real zeros.

1. List all of the possible rational zeros.

2. Use the graph to pick one that works.

3. Use synthetic division to factor the polynomial.

4. Now what you see is still the same function, only it has been factored. There
is the linear factor (x-c) where c is your rational zero. The other part is the
quotient from the synthetic division. It is called the de pressed equation .

5. To continue finding real zeros of f, we now try to find real zeros of the
depressed equation. The process is long and yes, depressing.

6. But, as the depressed equation is always a lower degree polynomial, we may
eventually be able to factor it by easier methods like the quadratic formula .
Also, it may not be able to be factored at all and we are done finding real zeros.

Examples:

f (x) = 2x3 −13x2 + 24x − 9

f (x) = 2x4 +17x 3 + 35x2 −9x − 45

f (x) = 4 x5 −8x4 − x + 2

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of December 13th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2017, Algebra-Answer.Com.  All rights reserved.