Variation and Polynomial Equations Chapter 8 Variation and Polynomial Equations

Direct Variation and Proportion 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 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.

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

Proportion An equality of ratios y1 = y2 x1 x2

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

Constant of Proportionality m is the constant of proportionality

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

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

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

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?

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

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

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

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

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

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?

Section 8-3 Dividing Polynomials

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

Example 1 Divide x3 – 5x2 + 4x – 2 x – 2 x2 – 3x – 2 + -6/x-2

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

Section 8-4 Synthetic Division

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

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

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

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

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.

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

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

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

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

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).

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