1. 5.3 The Fundamental Theorem of Algebra & Descartes’ Rule of Signs
Today’s Date:
12/18/17

2. Fundamental Theorem of Algebra
Any polynomial of degree n has n roots.
May be complex roots
May have duplicates (multiplicity)

3. Ex 1) State the degree & list distinct zeros & their multiplicities.
To find degree when in factored form, add exponents (total)
Degree =
1 + 2 = 3
Zeros:
1 (mult. 1)
–2 (mult. 2)
Parenthesis → Opp sign
Multiplicity → Exponent

4. T.O.O. Ex 2) State the degree & list distinct zeros & their multiplicities.
= 7
Zeros:
0 (mult. 2)
–3 (mult. 2)
¾ (mult. 3)

5. Ex 3)
Find poly. of least degree that is monic & zeros are 3 (mult. 2) and –1. Write in factored form.
Monic: lead. coeff. = 1

6. Find poly. of least degree with zeros 1, –1, and 2 and P(0) = 4.
Ex 4)
Find poly. of least degree with zeros 1, –1, and 2 and P(0) = 4.
Given to find leading coeff
Plug in 0 for x and set = to 4 to find a

7. Descartes’ Rule of Signs
Determines nature of the roots
~ the # of (+) or (–) real roots
~by counting # of sign changes

8. Ex 5) Use Descartes’ Rule of Signs to discuss the nature of the roots+
Count sign changes
change
change
# of (+) real roots: # of sign changes, or less by some multiple of 2 (SUBTRACT BY 2)
(+) Real Roots: 2 sign changes
→ 2 or 0 (+) real roots
(SUBTRACT BY 2)

9. Ex 5 cont.. (–) Real Roots: 1 sign change → 1 (–) real root
To find (–) real roots: Plug in (–x)
Count sign changes
change
(–) Real Roots: 1 sign change
→ 1 (–) real root
Hint: Only odd exponents will change sign when plugging in –x
Not or 0! Why?

10. Example 5 cont… Summarize your results in table form (+) (–) i 2 1 1 2
2 or 0 (+) real roots;
1 (–) real root
Summarize your results in table form
(+)
(–) i
Deg = 3 → 3 zeros
(each row must add to 3) 2 1 1 2
2 scenarios

11. T.O.O. Ex 6) Use Descartes’ Rule of Signs to discuss the nature of the roots
2 or 0 (+) real roots +
2 or 0 (–) real roots
(+)
(–) i 2 2
Deg = 4 → 4 zeros 2 2 2 2 4
4 scenarios

12. Homework
#505 Pg – 25 odd

13. Intermediate Value Theorem
If values of P(x) change from (+) to (–) or (–) to (+), there must be a zero in between.
P(x) is (+)
Zero in between
(it crosses the x-axis!)
P(x) is (–)

14. Ex 4) Use I.V.T. & synthetic division to show that P(x) has a zero between -3 & -4 ↓ -3 6 9
(+) 1 -2 -3 15
Changes signs, so must be a 0 between -4 ↓ -4 12
-12 1 -3 3 -6
(–)

15. FYI – the Real Graph of
Zoomed in to see zero