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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

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
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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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