Conjugate theorems, Descartes' rule of signs, Theorem on Upper bound and lower bound

1. Theorems on Polynomial Functions - Part 2
PSHS Main Campus
July 17, 2012
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 1 / 11

2. Review of Previous Discussion
1 Remainder Theorem
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11

3. Review of Previous Discussion
1 Remainder Theorem
2 Factor Theorem
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11

4. Review of Previous Discussion
1 Remainder Theorem
2 Factor Theorem
3 Rational Zeros/Roots Theorem
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11

5. Review of Previous Discussion
1 Remainder Theorem
2 Factor Theorem
3 Rational Zeros/Roots Theorem
4 Fundamental Theorem of Algebra
Example
Graph the function f (x) = 2x4 + 7x3 − 17x2 − 58x − 24.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 2 / 11

6. Quiz 7
Given the function f (x) = 3x4 + 14x3 + 16x2 + 2x − 3.
1 How many zeroes does f (x) have?
2 List all the possible rational zeroes according to RZT.
3 Express f (x) as a product of binomial factors.
4 Graph f (x), and correctly label all intercepts.
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7. Conjugate Theorems
Find the zeros of the following functions:
1 y = x3 − 4x − 15
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8. Conjugate Theorems
Find the zeros of the following functions:
1 y = x3 − 4x − 15
2 y = −x3 − 6x2 + 16
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9. Conjugate Theorems
Square root conjugates
Square Root Conjugate Theorem
Given a polynomial function f (x) with integer coefficients:
If:
√
1 (a + b c) is a zero of f (x),
2 a, b, c ∈ R, b, c = 0
√
then (a − b c) is also a zero of f (x).
Example
√
(−2 + 2 3) is a zero of −x3 − 6x2 + 16.
√
(−2 − 2 3) must also be a zero.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 5 / 11

10. Conjugate Theorems
Complex conjugates
Complex Conjugate Theorem
Given a polynomial function f (x) with real coefficients:
If:
1 (a + bi) is a zero of f (x),
2 a, b ∈ R, b = 0
then (a − bi) is also a zero of f (x).
Example
√
−3 + i 11
is a zero of x3 − 4x − 15.
2
√
−3 − i 11
must also be a zero.
2
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11. Theorem on Upper Bound and Lower Bound
Theorem on Upper Bound and Lower Bound
Suppose P (x) is a polynomial function divided by (x − r):
1 If r > 0 and all values on the quotient row are non-negative (positive
and zero), then r is an upper bound.
2 If r < 0 and the values on the quotient row are alternately
non-negative and non-positive, then r is a lower bound.
Example
Find the zeroes of f (x) = x3 − 6x2 + 5x + 12.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 7 / 11

12. Descartes’ Rule of Signs
Descartes’ Rule of Signs
Given polynomial function f (x) with real coefficients and non-zero
constant term:
The number of positive real zeros of f (x) is equal to the number of
variations of sign in f (x) or less than that by an even integer.
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13. Descartes’ Rule of Signs
Descartes’ Rule of Signs
Given polynomial function f (x) with real coefficients and non-zero
constant term:
The number of positive real zeros of f (x) is equal to the number of
variations of sign in f (x) or less than that by an even integer.
The number of negative real zeros of f (x) is equal to the number of
variations of sign in f (−x) or less than that by an even integer.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 8 / 11

14. Examples
Descartes’ Rule of Signs
Examples
1 How many positive and negative real zeroes does
f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have?
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15. Examples
Descartes’ Rule of Signs
Examples
1 How many positive and negative real zeroes does
f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have?
2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3)
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11

16. Examples
Descartes’ Rule of Signs
Examples
1 How many positive and negative real zeroes does
f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have?
2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3)
2 How many positive and negative real zeroes does
g(x) = 2x4 − 7x3 − 9x2 − 21x − 45 have?
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11

17. Examples
Descartes’ Rule of Signs
Examples
1 How many positive and negative real zeroes does
f (x) = 2x4 − 11x3 + 14x2 + 9x − 18 have?
2x4 − 11x3 + 14x2 + 9x − 18 = (x + 1)(x − 3)(x − 2)(2x − 3)
2 How many positive and negative real zeroes does
g(x) = 2x4 − 7x3 − 9x2 − 21x − 45 have?
2x4 − 7x3 − 9x2 − 21x − 45 = (2x + 3)(x − 5)(x2 + 3).
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 9 / 11

18. Exercises
1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11

19. Exercises
1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18.
2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11

20. Exercises
1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18.
2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.
3 How many negative zeroes does the function
f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x).
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11

21. Exercises
1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18.
2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.
3 How many negative zeroes does the function
f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x).
1 1 2
4 Solve for x: 1 − − 2 − 3 = 0.
x x x
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11

22. Exercises
1 Find the zeroes of y = 12x4 − 56x3 + 79x2 − 21x − 18.
2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.
3 How many negative zeroes does the function
f (x) = x4 − 10x3 + 35x2 − 50x + 24 have? Find all zeroes of f (x).
1 1 2
4 Solve for x: 1 − − 2 − 3 = 0.
x x x
5 Find the solution set of x in the inequality 2x3 + 5x2 − 14x − 8 ≤ 0.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 10 / 11

23. Exercises
1 Find the length of the edge of a cube if an increase of 3 cm in one
dimension and of 6 cm in another, and a decrease of 2 cm in the
third, doubles the volume.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11

24. Exercises
1 Find the length of the edge of a cube if an increase of 3 cm in one
dimension and of 6 cm in another, and a decrease of 2 cm in the
third, doubles the volume.
2 The dimensions of a block of metal are 3 cm, 4 cm, and 5 cm,
respectively. If each dimension is increased by the same number of
centimeters, the volume of the block becomes 3.5 times its original
volume. Determine how many centimeters were added to each
dimension.
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11

25. Exercises
1 Find the length of the edge of a cube if an increase of 3 cm in one
dimension and of 6 cm in another, and a decrease of 2 cm in the
third, doubles the volume.
2 The dimensions of a block of metal are 3 cm, 4 cm, and 5 cm,
respectively. If each dimension is increased by the same number of
centimeters, the volume of the block becomes 3.5 times its original
volume. Determine how many centimeters were added to each
dimension.
3 How long is the edge of a wooden cube if, after a slice of 1 cm thick
is cut off from one side, the volume of the remaining solid is 100
cubic cm?
PSHS Main Campus () Theorems on Polynomial Functions - Part 2 July 17, 2012 11 / 11