section 5.5 the real zeros of a polynomial function
TRANSCRIPT
Let f be a polynomial function. If we divide f (x)
by x c, then the remainder is f (c).
Remainder Theorem
3 2Find the remainder if 3 2 1 isdivided by
(a) 2 (b) 1
f x x x x
x x
3 2(a) 2 2 3 2 2 2 1 1f
The remainder is 1.
3 21(b) 1 1 1 3 2 1 5f
The remainder is 5.
Using the Remainder Theorem
3 2
Use the Factor Theorem to determine whether the function2 4 3 has the factor
(a) 1 (b) 1
f x x x x
x x
3 2(a) 1 2 1 1 4 1 3 0f
By the factor theorem, x + 1 is a factor of f (x).
3 2(b) 1 2 1 1 4 1 3 4f
By the factor theorem, x 1 is not a factor of f (x).
Using the Factor Theorem
3 2
List the potential rational zeros of3 8 7 12f x x x x
Factors of the constant
Factors of the leading coefficient
: 1, 2, 3, 4, 6, 12p : 1, 3q 1 2 4
: 1, 2, 3, 4, 6, 12, , ,3 3 3
p
q
Example
1 2 4: 1, 2, 3, 4, 6, 12, , ,
3 3 3
p
q
3 2Find the real zeros of the polynomial function 3 8 7 12.Write in factored form.
f x x x xf
Since f is a polynomial of degree 3, there are at most three real zeros.
21 3 5 12f x x x x
1 3 3 4x x x 1 3 3 4 0x x x
41 or 3 or
3x x x
4 3 2Find the real zeros of 2 13 29 27 9.Write in factored form.
f x x x x xf
Step 1: There are at most 4 real zeros.1 3 9
The potential rational zeros are : 1, , 3, , 9,2 2 2
p
q Step 2:
3 2 1 2 11 18 9f x x x x x
2 2 1 2 9 9f x x x x
2 1 3 2 3f x x x x
31 or 3 or
2x x x 2
1 3 2 3 0x x x
4 3 2Solve the equation: 2 13 29 27 9 0x x x x
21 2 3 3 0x x x
31, , 3
2
Solving a Polynomial Equation
The solution set is given by