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Algebra II CP1


									Algebra II CP1                 7.5 Zeros of Polynomial Functions

A polynomial is a FACTOR of another polynomial if __________________________________


Example: Use substitution or synthetic division to determine if x + 1 is a factor of
P( x)  3x3  4 x 2  x  5

Yesterday we found some roots using the calculator and long division. We will do the same today,
with some tougher answers

Example: Find all of the rational roots of 10 x3  9 x 2  19 x  6  0 .

Examine the graph of the related function P( x)  10 x 3  9 x 2  19 x  6 on your graphing calculator
and test all solutions using synthetic division or long division.

One root of P( x)  0 appears to be -2. Test whether P(2)  0 is true.

From the graph of the related function, there appears to be two real zeros or one real zero with
a multiplicity of 2 between 0 and 1. (“multiplicity of 2” refers to a factored situation like
( x  3)( x  3) where 3 is a solution twice)

A closer look between 0 and 1 on the graph of the related function shows 2 zeros…find them on
the calc and test them using synthetic division.

2nd possible root=                                    3rd possible root=
Test using synthetic division:

Factored Equation: ______________________________________

ALL the zeros are ______, ______, _______

*If a function is a cubic (power of 3) then it has at least one real zero. If it is a quartic (power
of 4) then it has at least 2 real zeros. Use this as your starting point when asked to find all
zeros of a polynomial function.

You can often use the   quadratic formula to solve a polynomial equation.

Example 2. Find all the zeros of Q( x)  x 3  2 x 2  2 x  1
Sometimes you will find that your equation only crosses the x-axis once, when you originally
thought it would cross three times (since it is a cubic). When this happens, it means that two of
your roots are IMAGINARY!!

Example 3. Find all the zeros of P( x)  3x 3  10 x 2  10 x  4

Now that you know how to find the roots, let’s work backwards. Let’s say that I GIVE you one or
two of the roots, and ask you to expand it to find the original polynomial.

Example 4: Find the Q(x) if two of the roots of the equation are -2 and 4 + i.

       First, find the OTHER complex root: _____________

       Now, write the equation in FACTORED form: Q(x) = _________________________

       Time for some distribution! I would distribute the two complex roots first, then
       distribute the real root. Be careful with your signs while multiplying!

       Q(x) =
Example 5. Find all the zeros of P( x)  x 3  6 x 2  7 x  2

Example 6. Find all the zeros of P( x)  x 3  9 x 2  49 x  145

Homework: page 464 #’s 23-33 odd (check back of the book for correctness…but show all

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