1. Section 11-7
The Rational-Zero Theorem
Mind your p’s and q’s
Tuesday, March 17, 2009
2. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
Tuesday, March 17, 2009
3. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
p = factors of 6:
Tuesday, March 17, 2009
4. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
p = factors of 6: ±1, ±2, ±3, ±6
Tuesday, March 17, 2009
5. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
p = factors of 6: ±1, ±2, ±3, ±6
q = factors of 2:
Tuesday, March 17, 2009
6. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
p = factors of 6: ±1, ±2, ±3, ±6
q = factors of 2: ±1, ±2
Tuesday, March 17, 2009
7. Warm-up
If p is a factor of 6 and q is a factor of 2, name all
possible fractions for p/q.
p = factors of 6: ±1, ±2, ±3, ±6
q = factors of 2: ±1, ±2
p 1 3
: ±1, ± ± 2, ±3, ± , ±6
q 2 2
Tuesday, March 17, 2009
9. Rational-Zero Theorem
Suppose all coefficients of the polynomial function
n n −1 2
P (x ) = a n x + a n −1x + ... + a2 x + a1x + a0
are integers with an ≠ 0 and a0 ≠ 0. Let p/q be a
rational number in lowest terms. If p/q is a zero of
P(x), then p is a factor of a0 and q is a factor of an.
Tuesday, March 17, 2009
10. Rational-Zero Theorem
Suppose all coefficients of the polynomial function
n n −1 2
P (x ) = a n x + a n −1x + ... + a2 x + a1x + a0
are integers with an ≠ 0 and a0 ≠ 0. Let p/q be a
rational number in lowest terms. If p/q is a zero of
P(x), then p is a factor of a0 and q is a factor of an.
What does that mean?
Tuesday, March 17, 2009
11. Rational-Zero Theorem
Suppose all coefficients of the polynomial function
n n −1 2
P (x ) = a n x + a n −1x + ... + a2 x + a1x + a0
are integers with an ≠ 0 and a0 ≠ 0. Let p/q be a
rational number in lowest terms. If p/q is a zero of
P(x), then p is a factor of a0 and q is a factor of an.
What does that mean?
We can find all possible roots of a polynomial by
examining factors of the leading coefficient and final
coefficient term (the one without a variable).
Tuesday, March 17, 2009
12. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Tuesday, March 17, 2009
13. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
Tuesday, March 17, 2009
14. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12:
Tuesday, March 17, 2009
15. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
Tuesday, March 17, 2009
16. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
q = factors of 3:
Tuesday, March 17, 2009
17. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
q = factors of 3: ±1, ±3
Tuesday, March 17, 2009
18. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
q = factors of 3: ±1, ±3
p 1 2 4
: ±1, ± , ±2, ± , ±3, ±4, ± , ±6, ±12
q 3 3 3
Tuesday, March 17, 2009
19. Example 1
Find the rational roots of
3 2
f (m) = 3m + 13m + 21m + 12
Find all possible roots.
p = factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
q = factors of 3: ±1, ±3
p 1 2 4
: ±1, ± , ±2, ± , ±3, ±4, ± , ±6, ±12
q 3 3 3
Calculator!
Tuesday, March 17, 2009
45. Using the Rational-Zero
Theorem
1. Find factors of p (last coefficient, all
by itself)
Tuesday, March 17, 2009
46. Using the Rational-Zero
Theorem
1. Find factors of p (last coefficient, all
by itself)
2. Find factors of q (leading coefficient)
Tuesday, March 17, 2009
47. Using the Rational-Zero
Theorem
1. Find factors of p (last coefficient, all
by itself)
2. Find factors of q (leading coefficient)
3. Determine all possibilities of p/q
Tuesday, March 17, 2009
48. Using the Rational-Zero
Theorem
1. Find factors of p (last coefficient, all
by itself)
2. Find factors of q (leading coefficient)
3. Determine all possibilities of p/q
4. Use this information to help you find
zeros in the calculator
Tuesday, March 17, 2009
