Chapter 8 Variation and Polynomial Equations. Section 8-1 Direct Variation and Proportion.
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Transcript of Chapter 8 Variation and Polynomial Equations. Section 8-1 Direct Variation and Proportion.
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Chapter 8Chapter 8
Variation and Variation and Polynomial Polynomial EquationsEquations
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Section 8-1Section 8-1
Direct Variation Direct Variation and Proportionand Proportion
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Direct VariationDirect Variation
A linear function A linear function defined by an defined by an equation of the form equation of the form y = mxy = mx
y varies directly as xy varies directly as x
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Constant Constant VariationVariation
The constant The constant mm is the is the constant variationconstant variation
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Example 1Example 1 The stretch is a loaded The stretch is a loaded spring varies directly as spring varies directly as the load it supports. A the load it supports. A load of 8 kg stretches a load of 8 kg stretches a certain spring 9.6 cm. certain spring 9.6 cm.
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Find the constant of Find the constant of variation (m) and the variation (m) and the equation of direct equation of direct variation.variation.
m = 1.2m = 1.2 y = 1.2xy = 1.2x What load would stretch What load would stretch the spring 6 cm?the spring 6 cm?
5 kg5 kg
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ProportionProportion An equality of ratiosAn equality of ratios
yy11 = y = y22
xx11 x x22
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Directly Directly ProportionalProportional
In a direct variation, y In a direct variation, y is said to be directly is said to be directly proportional to xproportional to x
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Constant of Constant of ProportionalityProportionality
m is the constant of m is the constant of proportionalityproportionality
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Means and Means and ExtremesExtremes
meansmeansyy11:x:x11 = y = y22:x:x22
extremes
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Solving a Solving a ProportionProportion
The product of the The product of the extremes equals the extremes equals the product of the meansproduct of the means
yy11xx22 = y = y22xx11
To get this product, To get this product, cross multiplycross multiply
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Example 2Example 2
If y varies directly as If y varies directly as x, and y = 15 when x, and y = 15 when x=24, find x when y = x=24, find x when y = 25.25.
x = 40x = 40
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Example 3Example 3 The electrical resistance The electrical resistance in ohms of a wire varies in ohms of a wire varies directly as its length. If a directly as its length. If a wire 110 cm long has a wire 110 cm long has a resistance of 7.5 ohms, resistance of 7.5 ohms, what length wire will have what length wire will have a resistance of 12 ohms?a resistance of 12 ohms?
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Section 8-2Section 8-2
Inverse and Inverse and Joint VariationJoint Variation
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Inverse VariationInverse VariationA function defined by A function defined by an equation of the form an equation of the form xy = kxy = k or or y = k/xy = k/x
y varies inversely as x, y varies inversely as x, or y is inversely or y is inversely proportional to xproportional to x
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Example 1Example 1
If y is inversely If y is inversely proportional to x, and proportional to x, and y = 6 when x = 5, find y = 6 when x = 5, find x when y = 12.x when y = 12.
x = 2.5x = 2.5
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Joint VariationJoint Variation
When a quantity When a quantity varies directly as the varies directly as the product of two or product of two or more other quantitiesmore other quantities
Also called Also called jointly jointly proportionalproportional
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Example 2Example 2 If z varies jointly as x If z varies jointly as x and the square root and the square root of y, and z = 6 when x of y, and z = 6 when x = 3 and y = 16, find z = 3 and y = 16, find z when x = 7 and y = 4.when x = 7 and y = 4.
z = 7z = 7
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Example 3Example 3 The time required to travel a The time required to travel a given distance is inversely given distance is inversely proportional to the speed of proportional to the speed of travel. If a trip can be made in travel. If a trip can be made in 3.6 h at a speed of 70 km/h, 3.6 h at a speed of 70 km/h, how long will it take to make how long will it take to make the same trip at 90 km/h?the same trip at 90 km/h?
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Section 8-3Section 8-3
Dividing Dividing PolynomialsPolynomials
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Long DivisionLong DivisionUse the long division Use the long division process for polynomialsprocess for polynomials
Remember:Remember:
873 ÷ 14 = ?873 ÷ 14 = ?62 5/1462 5/14
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Example 1Example 1
DivideDivide
xx33 – 5x – 5x22 + 4x – 2 + 4x – 2
x – 2x – 2xx22 – 3x – 2 + -6/x-2 – 3x – 2 + -6/x-2
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CheckCheck
To check use the To check use the algorithm:algorithm:
Dividend = (quotient)Dividend = (quotient)(divisor) + remainder(divisor) + remainder
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Section 8-4Section 8-4
Synthetic Synthetic DivisionDivision
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Synthetic DivisionSynthetic Division
An efficient way to An efficient way to divide a polynomial divide a polynomial by a binomial of the by a binomial of the form form x – cx – c
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Reminder:Reminder:
The divisor must be in The divisor must be in the form the form x – cx – c
If it is not given in If it is not given in that form, put it into that form, put it into that formthat form
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Example 1Example 1
Divide:Divide:
xx44 – 2x – 2x33 + 13x – 6 + 13x – 6
x + 2x + 2 xx33 – 4x – 4x22 + 8x - 3 + 8x - 3
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Section 8-5Section 8-5
The Remainder The Remainder and Factor and Factor TheoremsTheorems
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Remainder Remainder TheoremTheorem
Let P(x) be a polynomial Let P(x) be a polynomial of positive degree of positive degree n.n. Then for any number Then for any number cc, , P(x) = Q(x)(x – c) + P(c) P(x) = Q(x)(x – c) + P(c) where Q(x) is a where Q(x) is a polynomial of degree n-1.polynomial of degree n-1.
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Remainder Remainder TheoremTheorem
You can use synthetic You can use synthetic division as “synthetic division as “synthetic substitution” in order substitution” in order to evaluate any to evaluate any polynomialpolynomial
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Synthetic Synthetic SubstitutionSubstitution
Evaluate at P(-4)Evaluate at P(-4)
P(x) = xP(x) = x44 – 14x – 14x22 + 5x – 3 + 5x – 3 Use synthetic division Use synthetic division to find the remainder to find the remainder when c = -4 when c = -4
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Factor TheoremFactor Theorem
The polynomial P(x) The polynomial P(x) has has x – rx – r as a factor if as a factor if and only if and only if rr is a root is a root of the equation P(x) = of the equation P(x) = 00
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ExampleExampleDetermine whether Determine whether x + 1 is a factor of x + 1 is a factor of P(x) = xP(x) = x1212 – 3x – 3x88 – 4x – 2 – 4x – 2 If P(-1) = 0, then x + If P(-1) = 0, then x + 1 is a factor1 is a factor
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ExampleExampleFind a polynomial equation Find a polynomial equation with integral coefficients that with integral coefficients that has 1, -2 and 3/2 as rootshas 1, -2 and 3/2 as roots
The polynomial must have The polynomial must have factors (x – 1), (x – (-2)) and (x factors (x – 1), (x – (-2)) and (x – 3/2).– 3/2).
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Depressed Depressed EquationEquation
Solve xSolve x33 + x + 10 = 0, + x + 10 = 0, given that -2 is a rootgiven that -2 is a root
To find the solution, To find the solution, divide the polynomial divide the polynomial by x – (-2)by x – (-2)