Transcript f(x) - jmullenkhs

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Homework “Mini-Quiz”
Use the paper provided - 10 min. (NO TALKING!!)
Do NOT write the question – Answer Only!!
A polynomial function written with terms in descending degree is written in
_______ form.
In the function above, n is called ______________.
Describe how f(x) = -(x+2)3 - 1 would change g(x) = x3.
Given any polynomial function f(x) = anxn+..+a1x+a0, if an > 0 and n is ____,
then
and
Given any polynomial function f(x) = anxn+..+a1x+a0, if an ___ 0 and n is even,
then
and
Graph f(x) = (x-2)2(x+1)(x-3). Describe the end behavior using limits. (#21)
Find the zeros of f(x) = x3-25x algebraically. (Show your work) (#36)
If you finish before the timer sounds, start the warm-up. (NO TALKING!!)
Warm-up (5 min.)
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2)
Using only algebra, find a cubic function with the
given zeros: 3,-4,6
Use cubic regression to fit a curve through the four
points given in the table:
x
-2
1
4
7
y
2
5
9
26
2.4 Real Zeros of Polynomial Functions
• Divide polynomials using long division or
synthetic division
• Apply the Remainder and Factor Theorem
• Find upper and lower bounds for zeros of
polynomials
Do you recall?
In the long division shown, what are the
names for the values: 30, 4, 7, and 2?
7
4 30
28
2
Ex1 Use long division to divide f(x) by d(x), and
write a summary statement in polynomial form and
fraction form.
a) f(x) = x2 - 2x + 3; d(x) = x – 1
b) f(x) = x4 - 2x3 + 3x2 - 4x + 6;
d(x) = x2 + 2x - 1
Long Division and the Division
Algorithm
Let f(x) and d(x) be polynomials with the degree of f
greater than or equal to the degree of d, and d(x)  0.
Then there are unique polynomials q(x) and r(x),
called the quotient and remainder, such that
f(x) = d(x) q(x) + r(x)
where either r(x) = 0 or the degree of r is less than
the degree of d.
The remainder determines a factor
• Remainder Theorem - If a polynomial f(x) is
divided by x - k, then the remainder is r = f(k).
• Factor Theorem - A polynomial function f(x)
has a factor x - k if and only if f(k) = 0.
Ex 2 Use the remainder theorem to find the
remainder when f(x) = 2x2 - 3x + 1 is divided by:
a) x – 2
b) x + 4
c) x – 1
Fundamental Connections for Polynomial Functions
For a polynomial function f and a real number k,
the following statements are equivalent:
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x = k is a solution (or root) of the equation f(x) = 0.
k is a zero of the function f.
k is an x-intercept of the graph of y = f(x).
x - k is a factor of f(x).
Ex 3 Use synthetic division to divide
f(x) = x3 + 5x2 + 3x – 2 by:
a) x + 1
b) x - 2
Synthetic Division
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Express the polynomial in standard form.
Use the coefficient (including zero coefficients) for
synthetic division.
3. Find the zero of the divisor (x - k = 0)
4. Use this as the divisor in the synthetic division
5. Bring down the leading coefficient
6. Multiply by the “zero divisor”
7. Add this product to the next coefficient
8. Repeat steps 4 & 5 until all coefficients have been
used
9. The last coefficient is the remainder
10. The other coefficient are the coefficients for the
quotient polynomial when written in standard form.
Upper and Lower Bounds
Let f be a polynomial function of degree n > 1 with a
positive leading coefficient. Suppose f(x) is divided by
x – k using synthetic division.
• If k > 0 and every number in the last line is nonnegative
(positive or zero), then k is an upper bound for the real
zeros of f.
• If k < 0 and the numbers in the last line are alternately
nonnegative and nonpositive, then k is a lower bound
for the real zeros of f.
Ex 4 Use synthetic division to prove that the number k
is the upper or lower bound (as stated) for the real
zeros of the function f.
a)
k = 3 is an upper bound; f(x) = 2x3 – 4x2 + x – 2
b)
k = -1 is a lower bound; f(x) = 3x3 – 4x2 + x + 3
Searching for zeros
Ex 5 Show that all the zeros of f(x) = 2x3 – 3x2
– 4x + 6 lie within the interval [-7,7]. Find all
of the zeros.
Rational Zeros (Roots) Theorem
If a polynomial f(x) = anxn + an-1xn-1 + … + a1x + a0
has any rational roots, then they are of the form p/q
where q is a factor of an (the leading coefficient) and
p is a factor of a0 (the constant term).
Ex 6 List all the possible rational roots of
f(x) = 2x3 + 5x2 - 3x + 5
Tonight’s Assignment
p. 216 - 218 Ex 3-33 m. of 3, 39,
42, 51-60 m. of 3
Exit Ticket
• Find all the roots of f(x) from Ex 6 and
place in the turn in box before you
leave.
• Have a great day!! 
• Remember to study!!