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Chapter 8

Chapter 8. Variation and Polynomial Equations. Section 8-1. Direct Variation and Proportion. Direct Variation. A linear function defined by an equation of the form y = mx y varies directly as x. Constant Variation. The constant m is the constant variation. Example 1.

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Chapter 8

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  1. Chapter 8 Variation and Polynomial Equations

  2. 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. • m = 1.2 • 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. 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 – 2 + -6/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. 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)

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