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( x 4 1 ) ÷ ( x 4 )

255

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( 3 x 3 + 4 x 2 8 x + 2 ) ÷ ( x 3 )

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( 4 x 3 + 5 x 2 2 x + 7 ) ÷ ( x + 2 )

1

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For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

f ( x ) = 2 x 3 9 x 2 + 13 x 6 ;   x 1

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f ( x ) = 2 x 3 + x 2 5 x + 2 ;   x + 2

2 ,   1 ,   1 2

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f ( x ) = 3 x 3 + x 2 20 x + 12 ;   x + 3

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f ( x ) = 2 x 3 + 3 x 2 + x + 6 ; x + 2

2

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f ( x ) = 5 x 3 + 16 x 2 9 ; x 3

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x 3 + 3 x 2 + 4 x + 12 ; x + 3

3

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4 x 3 7 x + 3 ; x 1

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2 x 3 + 5 x 2 12 x 30 , 2 x + 5

5 2 ,   6 ,   6

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For the following exercises, use the Rational Zero Theorem to find all real zeros.

x 3 3 x 2 10 x + 24 = 0

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2 x 3 + 7 x 2 10 x 24 = 0

2 ,   4 ,   3 2

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x 3 + 2 x 2 9 x 18 = 0

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x 3 + 5 x 2 16 x 80 = 0

4 ,   4 ,   5

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x 3 3 x 2 25 x + 75 = 0

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2 x 3 3 x 2 32 x 15 = 0

5 ,   3 ,   1 2

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2 x 3 + x 2 7 x 6 = 0

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2 x 3 3 x 2 x + 1 = 0

1 2 ,   1 + 5 2 ,   1 5 2

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3 x 3 x 2 11 x 6 = 0

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2 x 3 5 x 2 + 9 x 9 = 0

3 2

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2 x 3 3 x 2 + 4 x + 3 = 0

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x 4 2 x 3 7 x 2 + 8 x + 12 = 0

2 ,   3 ,   1 ,   2

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x 4 + 2 x 3 9 x 2 2 x + 8 = 0

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4 x 4 + 4 x 3 25 x 2 x + 6 = 0

1 2 ,   1 2 ,   2 ,   3

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2 x 4 3 x 3 15 x 2 + 32 x 12 = 0

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x 4 + 2 x 3 4 x 2 10 x 5 = 0

1 ,   1 ,   5 ,   5

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8 x 4 + 26 x 3 + 39 x 2 + 26 x + 6

3 4 ,   1 2

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For the following exercises, find all complex solutions (real and non-real).

x 3 8 x 2 + 25 x 26 = 0

2 ,   3 + 2 i ,   3 2 i

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x 3 + 13 x 2 + 57 x + 85 = 0

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3 x 3 4 x 2 + 11 x + 10 = 0

2 3 ,   1 + 2 i ,   1 2 i

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x 4 + 2 x 3 + 22 x 2 + 50 x 75 = 0

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2 x 3 3 x 2 + 32 x + 17 = 0

1 2 ,   1 + 4 i ,   1 4 i

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Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

f ( x ) = x 4 x 2 1

1 positive, 1 negative

Graph of f(x)=x^4-x^2-1.
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f ( x ) = x 3 2 x 2 5 x + 6

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f ( x ) = x 3 2 x 2 + x 1

3 or 1 positive, 0 negative

Graph of f(x)=x^3-2x^2+x-1.
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f ( x ) = x 4 + 2 x 3 12 x 2 + 14 x 5

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f ( x ) = 2 x 3 + 37 x 2 + 200 x + 300

0 positive, 3 or 1 negative

Graph of f(x)=2x^3+37x^2+200x+300.
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f ( x ) = x 3 2 x 2 16 x + 32

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f ( x ) = 2 x 4 5 x 3 5 x 2 + 5 x + 3

2 or 0 positive, 2 or 0 negative

Graph of f(x)=2x^4-5x^3-5x^2+5x+3.
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f ( x ) = 2 x 4 5 x 3 14 x 2 + 20 x + 8

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f ( x ) = 10 x 4 21 x 2 + 11

2 or 0 positive, 2 or 0 negative

Graph of f(x)=10x^4-21x^2+11.
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Numeric

For the following exercises, list all possible rational zeros for the functions.

f ( x ) = x 4 + 3 x 3 4 x + 4

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f ( x ) = 2 x 3 + 3 x 2 8 x + 5

± 5 ,   ± 1 ,   ± 5 2

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f ( x ) = 3 x 3 + 5 x 2 5 x + 4

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f ( x ) = 6 x 4 10 x 2 + 13 x + 1

± 1 ,   ± 1 2 ,   ± 1 3 ,   ± 1 6

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f ( x ) = 4 x 5 10 x 4 + 8 x 3 + x 2 8

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Technology

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

f ( x ) = 6 x 3 7 x 2 + 1

1 ,   1 2 ,   1 3

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f ( x ) = 4 x 3 4 x 2 13 x 5

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f ( x ) = 8 x 3 6 x 2 23 x + 6

2 ,   1 4 ,   3 2

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f ( x ) = 12 x 4 + 55 x 3 + 12 x 2 117 x + 54

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f ( x ) = 16 x 4 24 x 3 + x 2 15 x + 25

5 4

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Extensions

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1, 3 and ( 2 , f ( 2 ) ) = ( 2 , 4 )

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Real roots: –1 (with multiplicity 2 and 1) and ( 2 , f ( 2 ) ) = ( 2 , 4 )

f ( x ) = 4 9 ( x 3 + x 2 x 1 )

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Real roots: –2, 1 2 (with multiplicity 2) and ( 3 , f ( 3 ) ) = ( 3 , 5 )

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Real roots: 1 2 , 0, 1 2 and ( 2 , f ( 2 ) ) = ( 2 , 6 )

f ( x ) = 1 5 ( 4 x 3 x )

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Real roots: –4, –1, 1, 4 and ( 2 , f ( 2 ) ) = ( 2 , 10 )

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Real-world applications

For the following exercises, find the dimensions of the box described.

The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

8 by 4 by 6 inches

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The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.

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The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

5.5 by 4.5 by 3.5 inches

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The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

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The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

8 by 5 by 3 inches

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For the following exercises, find the dimensions of the right circular cylinder described.

The radius is 3 inches more than the height. The volume is 16 π cubic meters.

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The height is one less than one half the radius. The volume is 72 π cubic meters.

Radius = 6 meters, Height = 2 meters

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The radius and height differ by one meter. The radius is larger and the volume is 48 π cubic meters.

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The radius and height differ by two meters. The height is greater and the volume is 28.125 π cubic meters.

Radius = 2.5 meters, Height = 4.5 meters

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The radius is 1 3 meter greater than the height. The volume is 98 9 π cubic meters.

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Questions & Answers

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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