Chapter 2 Notes
Transcript Chapter 2 Notes
HW: Pg. 175 #7-16
◦ Let n be a nonnegative integer and let a0, a1, a2,…,
an-1, an be real numbers with an≠0. The functions
f(x)=anxn + an-1xn-1+…+a2x2+a1x+a0
Is a polynomial function of degree n. The leading
coefficient is an.
f(x)=0 is a polynomial function.
*it has no degree or leading coefficient.
F(x) = 5x3-2x-3/4
G(x) = √(25x4+4x2)
H(x) = 4x-5+6x
F(x) = 0
F(x) = ax+b
Slope-Intercept form of a line:
Find an equation for the linear function f such
that f(-2) = 5 and f(3) = 6
The average rate of change of a function
y=f(x) between x=a and x=b, a≠b, is
Weehawken High School bought a $50,000
building and for tax purposes are depreciating it
$2000 per year over a 25-yr period using
What is the rate of change of the value of the
Write an equation for the value v(t) of the building
as a linear function of the time t since the
building was placed in services.
Evaluate v(0) and v(16)
Point of View
Polynomial of degree 1
Slant line with slope m and yintercept b
Function with constant nonzero
rate of change m, f is increasing if
m>0, decreasing if m<0
Sketch how to transform f(x)=x2 into:
If g(x) and h(x) and in the form f(x)=ax2+bx+c,
what do you notice about g(x) and h(x) when a is
a certain value (negative or positive)?
We want to find the axis of symmetry, which is
The graph of f is a parabola with vertex (x,y),
where x=-b/(2a). If a>0, the parabola opens
upward, and if a<0, it opens downward.
Any Quadratic Function f(x)=ax2+bx+c, can
be written in the vertex form:
Where (h,k) is your vertex
h=-b/(2a) and k=is the y
=3(x2+4x+() - () )+11
Factor 3 from the x term
Prepare to complete the square.
Complete the square.
Distribute the 3.
Point of View
Polynomial of degree ___
HW: Pg.189 #1-10
◦ a is the power, k is the constant of variation
Constant of Variation
F(x) = ∛x
What type of Polynomials are these functions?
(HINT: count the terms)
HW: Pg 203 #33-42e
Find local extrema and zeros for each
What do you notice about the limits of each
What do these zeros tell us about our graph?
F(x)=3x3 + 12x2 – 15x
H(x)=x2 + 3x2 – 16
G(x)=9x3 - 3x2 – 2x
K(x)=2x3 - 8x2 + 8x
F(x)=6x2 + 18x – 24
HW: Pg. 216 #1-6
F(x) = d(x)*q(x)+r(x)
F(x) and d(x) are polynomials where q(x) is
the quotient and r(x) is the remainder
Write (2x4+3x3-2)/(2x2+x+1) in fraction form
D(x)=x-k, degree is 1, so the remainder is a
Divide f(x)=3x2+7x-20 by:
◦ (a) x+2
If a polynomial f(x) is divided by x-k, then the remainder
Divide f(x)=3x2+7x-20 by:
◦ (a) x+2
If d(x)=x-k, where f(x)=(x-k)q(x) + r
Then we can evaluate the polynomial f(x) at x=k:
Now we can use this method to find both
remainders and quotients for division by x-k,
called synthetic division.
K becomes zero of divisor
3 | 2 -3 -5 -12
* Since the leading coefficient of
the dividend must be the leading
coefficient , copy the first “2” into
the first quotient position.
* Multiply the zero of the divisor
(3) by the most recent coefficient
of the quotient (2). Write the
product above the line under the
next term (-3).
* Add the next coefficient of the
dividend to the product just found
and record sum below the line in
the same column.
* Repeat the “multiply” and “add”
steps until the last row is
Suppose f is a polynomial function of degree
n1 of the form f(x)=anxn+…+a0
with every coefficient an integer. If x=p/q is
a rational zero of f, where p and q have no
common integer factors other than 1, then
◦ P is an integer factor of the constant coefficient a0,
◦ Q is an integer factor of the leading coefficient an.
Example: Find rational zeros of f(x)=x3-3x2+1
Potential Rational Zeros:
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.
Lets establish that all the real zeros of
f(x)=2x4-7x3-8x2+14x+8 must lie in the
Establish bounds for real zeros
Find the real zeros of a polynomial functions
by using the rational zeros theorem to find
potential rational zeros
Use synthetic division to see which potential
rational zeros are a real zero
Complete the factoring of f(x) by using
synthetic division again or factor.
In the 17th century, mathematicians extended
the definition of √(a) to include negative real
i =√(-1) is defined as a solution of (i )2 +1=0
For any negative real number √(a) = √|a|*i
a +bi , where a, b are real numbers
◦ a+bi is in standard form
Sum: (a+bi) +(c+di) = (a+c) + (b+d)i
Difference: (a+bi) – (c+di) = (a-c) + (b-d)I
EX: (a) (8 - 2i) + (5 + 4i)
(b) (4 – i) – (5 + 2i)
Z=(1/2)+(√3/2)i, find Z2
Z = a+bi = a – bi
When do we need to use conjugates?
Find all zeros:
f(x) = x4 + x3 + x2 + 3x - 6
HW: Pg. 234-235 #2-10e, 28-34e
Fundamental Theorem of Algebra – A polynomial
function of degree n has n complex zeros (real
Linear Factorization Theorem – If f(x) is a
polynomial function of degree n>0, then f(x) has
n linear factors and
F(x) = a(x-z1)(x-z2)…(x-zn)
Where a is the leading coefficient of f(x) and z1, z2, …, zn
are the complex zeros of the function.
X=k is a…
K is a
Factor of f(x):
Suppose that f(x) is a polynomial function
with real coefficients. If a+bi is a zero of f(x),
then the complex conjugate a-bi is also a
zero of f(x)
What can happen if the coefficients are not real?
Use substitution to verify that x=2i and x=-i are
zeros of f(x)=x2-ix+2. Are the conjugates of 2i and –i
also zeros of f(x)?
Use substitution to verify that x=i and x=1-i are
zeros of g(x)=x2-x+(1+i). Are the conjugates of i and
1-i also zeros of g(x)?
What conclusions can you draw from parts 1 and 2?
Do your results contradict the theorem about complex
Given that -3, 4, and 2-i are zeros, find the
Given 1, 1+2i, 1-i, find the polynomial:
The complex number z=1-2i is a zero of
f(x)=4x4+17x2+14x+65, find the remaining
zeros, and write it in its linear factorization.
HW: Pg. 246 #19-30
F and g are polynomial functions with g(x)≠0.
◦ R(x)=f(x)/g(x) is a rational function
◦ Find the domain of : f(x)=1/(x+2)
Find horizontal and vertical asymptotes of
Find asymptotes and intercepts of the
HW: Pg. 254 #7-17
X + 3/x = 4
2/(x-1) + x = 5
(2x)/(x-1) + 1/(x-3) = 2/(x2-4x+3)
3x/(x+2) + 2/(x-1) = 5/(x2+x-2)
(x-3)/x + 3/(x+2) + 6/(x2 +2x) = 0
Find the dimensions of the rectangle with
minimum perimeter if its area is 200 square
meters. Find the least perimeter:
A = 200
HW: Finish 2.9 WKSH
Determine the real number values of x that
cause the function to be zero, positive, or
(x+3)(x2+1)(x-4)2 > 0
(x+3)(x2+1)(x-4)2 ≥ 0
(x+3)(x2+1)(x-4)2 < 0
(x+3)(x2+1)(x-4)2 ≤ 0
Solve Graphically: x3-6x2≤2-8x
*Plug function into your calculator*
Section 2.9 #1-12 odd