October 13, 2015
Find the vertex using 2 different methods, one which is completing the square.
(No calculator)
f(x) = 2x2 - 4x + 6
Method 1
Method 2 (completing the square)
October 13, 2015
Find the vertex using 2 different methods, one which is completing the square.
(No calculator)
f(x) = 2x2 - 4x + 6
Method 1
x=
x=
-b
f(x) = 2(x2 - 2x
2a
) +6
f(x) = 2(x2 - 2x + 1 ) + 6 - 2
4
2(2)
f(x) = 2(x - 1)2 + 4
Vertex: (1, 4)
x=1
f(1) = 2(1)2 - 4(1) + 6
f(1) = 2 - 4 + 6
f(1) = -2 + 6
f(1) = 4
Method 2 (completing the square)
Vertex (1, 4)
October 13, 2015
October 13, 2015
October 13, 2015
(2.4) Real Zeros of Polynomial Functions
Objective: To learn about Long Division of Polynomials,
Remainder and Factor Theorems, Synthetic Division, Rational
Zeros Theorem.
Why: These topics help identify real zeros of polynomial
functions.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Divide using long division.
3x3 + 5x2 + 8x + 7
by
3x + 2
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Find the remainder when f(x)=x3 - 3x + 4 is divided by x + 2.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Remainder Theorem:
If a polynomial f(x) is divided by x-k, then the remainder
is r = f(k).
Find the remainder when f(x)=3x2+ 7x - 20 is divided by x + 4.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Q: What does it mean to get a remainder of 0?
Factor Theorem:
A polynomial function f(x) has a factor of x - k iff f(k)=0.
Is x - 3 a factor of x3- x2- x- 15?
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Synthetic Division
Divide 2x - 3x 3 - 5x - 122 by x - 3 using synthetic division.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Rational Zeros Theorem:
If there are rational zeros in the polynomial:
f(x) = a x + an
xn
+ ...n+ -a1
then they will be in the set
n - 1
p
q
where p = factors of the constant, a
and
0
q = factors of the leading term coefficient, a
0
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Find the zeros of
f(x) = 3x + 4x 3- 5x - 2
2
October 13, 2015
Find the zeros.
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
f(x) = 2x4 - 7x3 - 8x2 + 14x + 8
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Upper and Lower Bound Tests for Real Zeros
Suppose f(x) is divided by x - k using synthetic division.
1. If k≥ 0 and every number in the last line is nonnegative (pos. or
zero), then k is an upper bound for the real zeros of f.
2. If k ≤ 0 and the numbers in the last line alternate nonnegative
and nonpositive, then k is a lower bound for the real zeros of f.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
Prove that k = 5 is an upper bound for the real
zeros of f(x) = 2x3 - 5x2 - 5x - 1
Prove that k = -4 is a lower bound for the real
zeros of f(x) = 3x3 - x2 - 5x - 3
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
HW
(2.4)
(HR) Pg.205: 2, 3, 7, 11, 13, 15, 19, 21, 27, 33, 35, 37, 41, 49, 51, 53
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.
October 13, 2015
Obj: To learn about Long Division of Polynomials, Remainder and Factor
Theorems, Synthetic Division, Rational Zeros Theorem.