1. Variation and Polynomial Equations
Chapter 8
Variation and Polynomial Equations

2. Direct Variation and Proportion
Section 8-1
Direct Variation and Proportion

3. Direct Variation
A linear function defined by an equation of the form y = mx
y varies directly as x

4. Constant Variation
The constant m is the constant variation

5. Example 1
The stretch is a loaded spring varies directly as the load it supports. A load of 8 kg stretches a certain spring 9.6 cm.

6. Find the constant of variation (m) and the equation of direct variation.
y = 1.2x
What load would stretch the spring 6 cm?
5 kg

7. Proportion
An equality of ratios
y1 = y2
x1 x2

8. Directly Proportional
In a direct variation, y is said to be directly proportional to x

9. Constant of Proportionality
m is the constant of proportionality

10. Means and Extremes
means
y1:x1 = y2:x2
extremes

11. Solving a Proportion
The product of the extremes equals the product of the means
y1x2 = y2x1
To get this product, cross multiply

12. Example 2
If y varies directly as x, and y = 15 when x=24, find x when y = 25.
x = 40

13. Example 3
The electrical resistance in ohms of a wire varies directly as its length. If a wire 110 cm long has a resistance of 7.5 ohms, what length wire will have a resistance of 12 ohms?

14. Inverse and Joint Variation
Section 8-2
Inverse and Joint Variation

15. Inverse Variation
A function defined by an equation of the form xy = k or y = k/x
y varies inversely as x, or y is inversely proportional to x

16. Example 1
If y is inversely proportional to x, and y = 6 when x = 5, find x when y = 12.
x = 2.5

17. Joint Variation
When a quantity varies directly as the product of two or more other quantities
Also called jointly proportional

18. Example 2
If z varies jointly as x and the square root of y, and z = 6 when x = 3 and y = 16, find z when x = 7 and y = 4.
z = 7

19. Example 3
The time required to travel a given distance is inversely proportional to the speed of travel. If a trip can be made in 3.6 h at a speed of 70 km/h, how long will it take to make the same trip at 90 km/h?

20. Section 8-3
Dividing Polynomials

21. Long Division Use the long division process for polynomials Remember:
873 ÷ 14 = ?
62 5/14

22. Example 1
Divide
x3 – 5x2 + 4x – 2
x – 2
x2 – 3x – /x-2

23. Check To check use the algorithm:
Dividend = (quotient)(divisor) + remainder

24. Section 8-4
Synthetic Division

25. Synthetic Division
An efficient way to divide a polynomial by a binomial of the form x – c

26. Reminder: The divisor must be in the form x – c
If it is not given in that form, put it into that form

27. Example 1
Divide:
x4 – 2x3 + 13x – 6
x + 2
x3 – 4x2 + 8x - 3

28. The Remainder and Factor Theorems
Section 8-5
The Remainder and Factor Theorems

29. Remainder Theorem
Let P(x) be a polynomial of positive degree n. Then for any number c, P(x) = Q(x)(x – c) + P(c) where Q(x) is a polynomial of degree n-1.

30. Remainder Theorem
You can use synthetic division as “synthetic substitution” in order to evaluate any polynomial

31. Synthetic Substitution
Evaluate at P(-4)
P(x) = x4 – 14x2 + 5x – 3
Use synthetic division to find the remainder when c = -4

32. Factor Theorem
The polynomial P(x) has x – r as a factor if and only if r is a root of the equation P(x) = 0

33. Example Determine whether x + 1 is a factor of
P(x) = x12 – 3x8 – 4x – 2
If P(-1) = 0, then x + 1 is a factor

34. Example
Find a polynomial equation with integral coefficients that has 1, -2 and 3/2 as roots
The polynomial must have factors (x – 1), (x – (-2)) and (x – 3/2).

35. Depressed Equation Solve x3 + x + 10 = 0, given that -2 is a root
To find the solution, divide the polynomial by x – (-2)