Evaluate the expression when x = –4
1. x2 + 5x
(-4)2 + 5(-4)
16 – 20
-4
2. –3x3 – 2x2 + 10
–3(-4)3 – 2(-4)2 + 10
192 – 32 + 10
170
Check your HW
EXAMPLE 1
Identify polynomial functions
Decide whether the function is a polynomial function.
If so, write it in standard form and state its degree, type, and leading
coefficient. Ask yourself – are the exponents all whole numbers???
Are the coefficients real.
1.
h (x) = x4 – 1 x2 + 3
4
3.
f (x) = 5x2 + 3x –1 – x
Yes, in standard form, deg 4, type quartic,
LC 1
No
GUIDED PRACTICE
for Examples 1
Decide whether the function is a polynomial function.
If so, write it in standard form and state its degree,
type, and leading coefficient.
2.
g (x) = 7x –
3 + πx2
ANSWER
Yes, g ( x)  x2  7 x  3
Deg: 2, type: quadratic, LC: π
4.
k (x) = x + 2x – 0.6x5
ANSWER
No
EXAMPLE 2
Evaluate by direct substitution
Use direct substitution to evaluate
5.
f (x) = 2x4 – 5x3 – 4x + 8 when x = 3.
f (x) = 2x4 – 5x3 – 4x + 8
Write original function.
f (3) = 2(3)4 – 5(3)3 – 4(3) + 8
Substitute 3 for x.
= 162 – 135 – 12 + 8
Evaluate powers and multiply.
= 23
Simplify
GUIDED PRACTICE
for Example 2
Use direct substitution to evaluate the polynomial
function for the given value of x.
6.
f (x) = x4 + 2x3 + 3x2 – 7; x = –2
f (x) = (-2)4 + 2(-2)3 + 3(-2)2 – 7
f (x) = 16 – 16 + 12 – 7
f (x) = 5
EXAMPLE 3
Evaluate by synthetic substitution
Use synthetic substitution to evaluate f (x) from
Example 2 when x = 3.
7.
f (x) = 2x4 – 5x3 – 4x + 8
STEP 1
STEP 2
STEP 3
Write the coefficients of f (x) in order of
descending exponents. Write the value at
which f (x) is being evaluated to the left.
9 15 The last number is the
6 3
answer f (3) = 23
2
1 3
5 23
Bring down the first coefficient(leading)
Multiply by the x-value and add to the
next coefficient – continue to the end
)
Example 4 (not on paper)
Use synthetic substitution to evaluate the
polynomial function for the given value of x.
6.
f(x) = 5x3 + 3x2 – x + 7; x = 2
2
5
5
3
-1
7
10
26
50
13
25
57
The answer is 57
for Examples 3 and 4
GUIDED PRACTICE
Use synthetic substitution to evaluate the
polynomial function for the given value of x.
8. f (x) = x4 + 2x3 + 3x2 -7; x = -2
-2
1
1
2
3
0
-7
-2
0
-6
12
0
3
-6
5
The answer is 5
Polynomial End Behavior Even Functions
Even Functions
Positive
Leading
Coefficient
Left: f(x)
+∞ as x
-∞ Right: f(x)
+∞ as x
+∞
Even Functions
Negative
Leading
Coefficient
Left: f(x) -∞ as x
-∞
Right: f(x)
-∞ as x
+∞
Polynomial End Behavior Odd Functions
Odd Functions
Positive
Leading
Coefficient
Left: f(x)
-∞ as x
-∞
Right: f(x)
+∞ as x
+∞
-∞
Right: f(x)
-∞ as x
+∞
Odd Functions
Negative
Leading
Coefficient
Left: f(x)
+∞ as x
EXAMPLE 4
Standardized Test Practice
ANSWER The correct answer is D.
GUIDED PRACTICE
for Examples 3 and 4
10. Describe the degree and leading coefficient and
end behavior of the polynomial function whose graph
is shown.
ANSWER
degree: odd, leading
coefficient: negative
Left: f(x) +∞ as x -∞
Right: f(x)
-∞ as x
+∞
GUIDED PRACTICE
for Examples 5 and 6
Graph the polynomial function.
2. f(x) = x4 – x3 – 4x2 + 4
x
-3
-2
f(x)
76
12
-1
0
1
2
4
0
2
3
-4
22
Classwork
Worksheet 5.2 and 5.2.2 finish all in class
Textbook Homework
Pages 341 – 342 (3- 48) multiples of 3