Transcript Welcome: [glorybethbecker.weebly.com]

Welcome:
• Pass out Zeros of Higher Polynomials WKS
HW Key: p. 212: 60, 62 (even)
Intermediate Value Theorem
Objectives & HW:
• The students will be able to apply the
Intermediate Value Theorem in determining if
there is a zero between two given values of x
and in approximating that zero.
• HW: Intermediate Value Theorem Problems
(back of notesheet)
3.6 Topics on the Theory of Polynomial Functions (1)
The Intermediate Value Theorem
The Intermediate Value Theorem for Polynomials
Let P(x) be a polynomial function with real
coefficients. If P(a) and P(b) have opposite signs,
then there is at least one value of c between a
and b for which P(c) = 0. Equivalently, the
equation P(x) = 0 has at least one real root
between a and b.
3.6 Topics on the Theory of Polynomial Functions (1)
EXAMPLE:
Approximating a Real Zero
a. Show that the polynomial function f(x) = x3 - 2x - 5 has a real zero
between 2 and 3.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
a. Let us evaluate f(x) at 2 and 3. If f(2) and f(3) have opposite signs, then
there is a real zero between 2 and 3. Using f(x) = x3 - 2x - 5, we obtain
f(2) = 23 - 2 * 2 - 5 = 8 - 4 - 5 = -1
and
f(3) = 33 - 2 * 3 - 5 = 27 - 6 - 5 = 16.
f (2) is negative.
f(3) is positive.
This sign change shows that the polynomial function has a real zero
between 2 and 3.
more
3.6 Topics on the Theory of Polynomial Functions (1)
EXAMPLE:
Approximating a Real Zero (cont.)
a. Show that the polynomial function f(x) = x3 - 2x - 5 has a real zero
between 2 and 3.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
b. A numerical approach is to evaluate f at successive tenths between 2 and
3, looking for a sign change. This sign change will place the real zero
between a pair of successive tenths. Use the table function (2nd Window)
of your calculator and set TblStart to 2 and ΔTbl to .1. Look at the table
values (2nd Graph) and scroll down until you see a sign change.
X
Y1
2
-1
2.1
0.061
Sign change
Sign change
The sign change indicates that f has a real zero between 2 and 2.1.
more
3.6 Topics on the Theory of Polynomial Functions (1)
EXAMPLE:
Approximating a Real Zero (cont.)
a. Show that the polynomial function f(x) = x3 - 2x - 5 has a real zero
between 2 and 3.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
b. We now follow a similar procedure to locate the real zero between
successive hundredths. Use the table function (2nd Window) of your
calculator and set TblStart to 2 and ΔTbl to .01. Look at the table values
(2nd Graph) and scroll down until you see a sign change.
f (2.00) = -1
f (2.04) = -0.590336
f (2.08) = -0.161088
f (2.01) = -0.899399
f (2.05) = -0.484875
f (2.09) = -0.050671
f (2.02) = -0.797592
f (2.06) = -0.378184
f (2.1) = 0.061
f (2.03) = -0.694573
f (2.07) = -0.270257
Sign change
The sign change indicates that f has a real zero between 2.09 and 2.1.
Correct to the nearest tenth, the zero is 2.1.
3.6 Topics on the Theory of Polynomial Functions (1)
You Try!!!
Approximating a Real Zero
a. Show that the polynomial function f(x) = 3x2 - 2x - 6 has a real zero
between 1 and 2.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
a. Let us evaluate f(x) at 1 and 2. If f(1) and f(2) have opposite signs, then
there is a real zero between 1 and 2. Using f(x) = 3x3 - 2x - 5, we obtain
f(1) = 3*12 - 2 * 1 - 6 = 3 - 2 - 6 = -5
and
f(2) = 3*22 - 2 * 2 - 6 = 12 - 4 - 6 = 2.
f (1) is negative.
f(2) is positive.
This sign change shows that the polynomial function has a real zero
between 1 and 2.
more
3.6 Topics on the Theory of Polynomial Functions (1)
EXAMPLE:
Approximating a Real Zero (cont.)
a. Show that the polynomial function f(x) = 3x2 - 2x - 6 has a real zero
between 1 and 2.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
b. A numerical approach is to evaluate f at successive tenths between 1 and
2, looking for a sign change. This sign change will place the real zero
between a pair of successive tenths. Use the table function (2nd Window)
of your calculator and set TblStart to 1 and ΔTbl to .1. Look at the table
values (2nd Graph) and scroll down until you see a sign change.
X
Y1
1.7
-0.73
1.8
0.12
Sign change
Sign change
The sign change indicates that f has a real zero between 1.7 and 1.8.
more
3.6 Topics on the Theory of Polynomial Functions (1)
EXAMPLE:
Approximating a Real Zero (cont.)
a. Show that the polynomial function f(x) = 3x2 - 2x - 6 has a real zero
between 1 and 2.
b. Use the Intermediate Value Theorem to find an approximation for this
real zero to the nearest tenth.
Solution
b. We now follow a similar procedure to locate the real zero between
successive hundredths. Use the table function (2nd Window) of your
calculator and set TblStart to 1 and ΔTbl to .01. Look at the table values
(2nd Graph) and scroll down until you see a sign change.
f (1.70) = -0.73
f (1.74) = -0.3972
f (1.78) = -0.0548
f (1.71) = -0.6477
f (1.75) = -0.3125
f (1.79) = 0.0323
f (1.72) = -0.5648
f (1.76) = -0.2272
f (1.80) = 0.12
f (1.73) = -0.4813
f (1.77) = -0.14.13
Sign change
The sign change indicates that f has a real zero between 1.78 and 1.79.
Correct to the nearest tenth, the zero is 1.8.
• You may start on your HW. (back of notesheet)