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Theorems on Polynomial Functions - Part 2

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July 17, 2012

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

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.

1 Remainder Theorem

2 Factor Theorem

Example

1 Remainder Theorem

2 Factor Theorem

Example

1 Remainder Theorem

2 Factor Theorem

Example

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.

Conjugate Theorems

Find the zeros of the following functions:

1 y = x3 − 4x− 15

2 y = −x3 − 6x2 + 16

Conjugate Theorems

Find the zeros of the following functions:

1 y = x3 − 4x− 15

2 y = −x3 − 6x2 + 16

Conjugate TheoremsSquare root conjugates

Square Root Conjugate Theorem

Given a polynomial function f(x) with integer coefficients:

1 (a+ b√c) is a zero of f(x),

2 a, b, c ∈ R, b, c 6= 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.

Conjugate TheoremsComplex conjugates

Complex Conjugate Theorem

Given a polynomial function f(x) with real coefficients:

1 (a+ bi) is a zero of f(x),

2 a, b ∈ R, b 6= 0

then (a− bi) is also a zero of f(x).

Example

−3 + i√11

2is a zero of x3 − 4x− 15.

−3− i√11

2must also be a zero.

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 (positiveand zero), then r is an upper bound.

2 If r < 0 and the values on the quotient row are alternatelynon-negative and non-positive, then r is a lower bound.

Example

Find the zeroes of f(x) = x3 − 6x2 + 5x+ 12.

Descartes’ Rule of Signs

Given polynomial function f(x) with real coefficients and non-zeroconstant term:

The number of positive real zeros of f(x) is equal to the number ofvariations 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 ofvariations of sign in f(−x) or less than that by an even integer.

Descartes’ Rule of Signs

Given polynomial function f(x) with real coefficients and non-zeroconstant term:

The number of positive real zeros of f(x) is equal to the number ofvariations 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 ofvariations of sign in f(−x) or less than that by an even integer.

ExamplesDescartes’ Rule of Signs

Examples

1 How many positive and negative real zeroes doesf(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 doesg(x) = 2x4 − 7x3 − 9x2 − 21x− 45 have?2x4 − 7x3 − 9x2 − 21x− 45 = (2x+ 3)(x− 5)(x2 + 3).

Examples

1 How many positive and negative real zeroes doesf(x) = 2x4 − 11x3 + 14x2 + 9x− 18 have?2x4 − 11x3 + 14x2 + 9x− 18 = (x+ 1)(x− 3)(x− 2)(2x− 3)

Examples

2 How many positive and negative real zeroes doesg(x) = 2x4 − 7x3 − 9x2 − 21x− 45 have?

2x4 − 7x3 − 9x2 − 21x− 45 = (2x+ 3)(x− 5)(x2 + 3).

Examples

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 functionf(x) = x4 − 10x3 + 35x2 − 50x+ 24 have? Find all zeroes of f(x).

4 Solve for x: 1− 1

x− 1

x2− 2

x3= 0.

5 Find the solution set of x in the inequality 2x3 + 5x2 − 14x− 8 ≤ 0.

Exercises

2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.

x− 1

x2− 2

x3= 0.

Exercises

2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.

x− 1

x2− 2

x3= 0.

Exercises

2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.

x− 1

x2− 2

x3= 0.

Exercises

2 Factor 6x5 + x4 − 9x3 + 14x2 − 24x.

x− 1

x2− 2

x3= 0.

Exercises

1 Find the length of the edge of a cube if an increase of 3 cm in onedimension and of 6 cm in another, and a decrease of 2 cm in thethird, 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 ofcentimeters, the volume of the block becomes 3.5 times its originalvolume. Determine how many centimeters were added to eachdimension.

3 How long is the edge of a wooden cube if, after a slice of 1 cm thickis cut off from one side, the volume of the remaining solid is 100cubic cm?

Exercises

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