Exam 6: Rational Expression and Equations

To quickly find the sum of a list of fractions, such as $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \frac { 1 } { 32 } + \frac { 1 } { 64 } + \frac { 1 } { 128 }$ mathematicians use the formula $S = \frac { b \left( 1 - r ^ { k } \right) } { 1 - r }$ Solve the formula for b .

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Question 1

Correct Answer:

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$b = \frac { S ( 1 - r ) } { 1 - r ^ { k } }$

a. What numbers cannot be solutions of the equation $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$ ? b. Solve: $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$

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Question 2

Correct Answer:

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a. $x = 0$ , $x = 5$
b. $\left\{ \frac { 5 } { 2 } , 10 \right\}$

Perform the operations and simplify the result when possible. $\frac { 5 x } { x ^ { 2 } + 11 x + 18 } + \frac { 5 x } { x ^ { 2 } - 4 }$

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Question 3

Correct Answer:

Verified

$\frac { 5 x \cdot ( 2 x + 7 ) } { ( x + 9 ) \cdot ( x - 2 ) \cdot ( x + 2 ) }$

Multiply, then simplify if possible. $\frac { x + 3 } { 7 } \cdot \frac { x } { x + 3 }$

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Question 4

Use synthetic division to perform the division. $\left( x ^ { 2 } - 9 x + 18 \right) \div ( x - 3 )$

(Multiple Choice)

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Question 5

A recipe for guacamole dip calls for 7 avocados. If they are advertised at 3 for $1.86, what will 7 avocados cost? $ __________

(Short Answer)

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Question 6

Add, and then simplify, if possible. $\frac { 7 } { x + 2 } + \frac { 8 } { x - 4 }$

(Multiple Choice)

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Question 7

Perform the division. $\frac { 6 a ^ { 3 } + a ^ { 2 } - 5 a + 7 } { a + 1 }$

(Multiple Choice)

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Question 8

Express the verbal model in symbols. C varies jointly with v , z , and x .

(Multiple Choice)

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Question 9

Use synthetic division to perform the division. $\frac { 3 - 3 x ^ { 2 } + 2 x } { x - 4 }$

(Multiple Choice)

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Question 10

A man roller-blades at a rate 6 miles per hour faster than he jogs. In the same time it takes him to roller-blade 6 miles he can jog 2 miles. How fast does he jog?

(Short Answer)

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Question 11

Perform the operations and simplify the result when possible. $\frac { x + 4 } { x + 5 } - \frac { x - 5 } { x + 7 }$

(Multiple Choice)

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Question 12

Find 3⁵ by using synthetic division to evaluate the polynomial $f ( x ) = x ^ { 5 } \text { at } x = 3$

(Short Answer)

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Question 13

Use synthetic division to perform the division. $\frac { 1 - 3 x ^ { 2 } + 2 x } { x - 4 }$

(Essay)

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Question 14

Divide using long division: $\frac { 2 x ^ { 3 } + 5 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 2 }$

(Multiple Choice)

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Question 15

Divide $20 x ^ { 2 } - 7 x - 2$ by $4 x + 5$ .

(Essay)

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Question 16

Multiply, then simplify if possible. $\frac { ( x + 3 ) ^ { 2 } } { x + 8 } \cdot \frac { x + 8 } { x + 3 }$

(Essay)

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Question 17

If P ( x ) is a polynomial and if $P ( k ) = 0$ , then k is called a __________ of the polynomial.

(Multiple Choice)

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Question 18

A boat that travels 19 mph in still water can travel 24 miles downstream in the same time as it takes to travel 14 miles upstream. Find the speed of the current in the river. __________ mph

(Short Answer)

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Question 19

Use the remainder theorem and synthetic division to find P ( k ). $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x - 5 ; k = 2$ $P ( 2 ) =$ __________

(Short Answer)

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Question 20

a. What numbers cannot be solutions of the equation $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$ ? b. Solve: $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$

Perform the operations and simplify the result when possible. $\frac { 5 x } { x ^ { 2 } + 11 x + 18 } + \frac { 5 x } { x ^ { 2 } - 4 }$

Multiply, then simplify if possible. $\frac { x + 3 } { 7 } \cdot \frac { x } { x + 3 }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 9 x + 18 \right) \div ( x - 3 )$

A recipe for guacamole dip calls for 7 avocados. If they are advertised at 3 for $1.86, what will 7 avocados cost? $ __________

Add, and then simplify, if possible. $\frac { 7 } { x + 2 } + \frac { 8 } { x - 4 }$

Perform the division. $\frac { 6 a ^ { 3 } + a ^ { 2 } - 5 a + 7 } { a + 1 }$

Express the verbal model in symbols. C varies jointly with v , z , and x .

Use synthetic division to perform the division. $\frac { 3 - 3 x ^ { 2 } + 2 x } { x - 4 }$

A man roller-blades at a rate 6 miles per hour faster than he jogs. In the same time it takes him to roller-blade 6 miles he can jog 2 miles. How fast does he jog?

Perform the operations and simplify the result when possible. $\frac { x + 4 } { x + 5 } - \frac { x - 5 } { x + 7 }$

Find 3⁵ by using synthetic division to evaluate the polynomial $f ( x ) = x ^ { 5 } \text { at } x = 3$

Use synthetic division to perform the division. $\frac { 1 - 3 x ^ { 2 } + 2 x } { x - 4 }$

Divide using long division: $\frac { 2 x ^ { 3 } + 5 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 2 }$

Divide $20 x ^ { 2 } - 7 x - 2$ by $4 x + 5$ .

Multiply, then simplify if possible. $\frac { ( x + 3 ) ^ { 2 } } { x + 8 } \cdot \frac { x + 8 } { x + 3 }$

If P ( x ) is a polynomial and if $P ( k ) = 0$ , then k is called a __________ of the polynomial.

A boat that travels 19 mph in still water can travel 24 miles downstream in the same time as it takes to travel 14 miles upstream. Find the speed of the current in the river. __________ mph

Use the remainder theorem and synthetic division to find P ( k ). $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x - 5 ; k = 2$ $P ( 2 ) =$ __________

A Review of Basic Algebra

Graphs, Equations of Lines, and Functions

Systems of Equations

Inequalities

Exponents, Polynomials, and Polynomial Functions

Radical Expressions and Equations

Quadratic Equations, Functions, and Inequalities

Exponential and Logarithmic Functions

Conic Sections; More Graphing

Miscellaneous Topics

Filters

Exam 6: Rational Expression and Equations

a. What numbers cannot be solutions of the equation xx−5+10x=3\frac { x } { x - 5 } + \frac { 10 } { x } = 3x−5x​+x10​=3 ? b. Solve: xx−5+10x=3\frac { x } { x - 5 } + \frac { 10 } { x } = 3x−5x​+x10​=3

Perform the operations and simplify the result when possible. 5xx2+11x+18+5xx2−4\frac { 5 x } { x ^ { 2 } + 11 x + 18 } + \frac { 5 x } { x ^ { 2 } - 4 }x2+11x+185x​+x2−45x​

Multiply, then simplify if possible. x+37⋅xx+3\frac { x + 3 } { 7 } \cdot \frac { x } { x + 3 }7x+3​⋅x+3x​

Use synthetic division to perform the division. (x2−9x+18)÷(x−3)\left( x ^ { 2 } - 9 x + 18 \right) \div ( x - 3 )(x2−9x+18)÷(x−3)

A recipe for guacamole dip calls for 7 avocados. If they are advertised at 3 for $1.86, what will 7 avocados cost? $ __________

Add, and then simplify, if possible. 7x+2+8x−4\frac { 7 } { x + 2 } + \frac { 8 } { x - 4 }x+27​+x−48​

Perform the division. 6a3+a2−5a+7a+1\frac { 6 a ^ { 3 } + a ^ { 2 } - 5 a + 7 } { a + 1 }a+16a3+a2−5a+7​

Express the verbal model in symbols. C varies jointly with v , z , and x .

Use synthetic division to perform the division. 3−3x2+2xx−4\frac { 3 - 3 x ^ { 2 } + 2 x } { x - 4 }x−43−3x2+2x​

A man roller-blades at a rate 6 miles per hour faster than he jogs. In the same time it takes him to roller-blade 6 miles he can jog 2 miles. How fast does he jog?

Perform the operations and simplify the result when possible. x+4x+5−x−5x+7\frac { x + 4 } { x + 5 } - \frac { x - 5 } { x + 7 }x+5x+4​−x+7x−5​

Find 35 by using synthetic division to evaluate the polynomial f(x)=x5 at x=3f ( x ) = x ^ { 5 } \text { at } x = 3f(x)=x5 at x=3

Use synthetic division to perform the division. 1−3x2+2xx−4\frac { 1 - 3 x ^ { 2 } + 2 x } { x - 4 }x−41−3x2+2x​

Divide using long division: 2x3+5x2−1x2−4x+2\frac { 2 x ^ { 3 } + 5 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 2 }x2−4x+22x3+5x2−1​

Divide 20x2−7x−220 x ^ { 2 } - 7 x - 220x2−7x−2 by 4x+54 x + 54x+5 .

Multiply, then simplify if possible. (x+3)2x+8⋅x+8x+3\frac { ( x + 3 ) ^ { 2 } } { x + 8 } \cdot \frac { x + 8 } { x + 3 }x+8(x+3)2​⋅x+3x+8​

If P ( x ) is a polynomial and if P(k)=0P ( k ) = 0P(k)=0 , then k is called a __________ of the polynomial.

A boat that travels 19 mph in still water can travel 24 miles downstream in the same time as it takes to travel 14 miles upstream. Find the speed of the current in the river. __________ mph

Use the remainder theorem and synthetic division to find P ( k ). P(x)=x3−3x2+x−5;k=2P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x - 5 ; k = 2P(x)=x3−3x2+x−5;k=2 P(2)=P ( 2 ) =P(2)= __________

A Review of Basic Algebra

Graphs, Equations of Lines, and Functions

Systems of Equations

Inequalities

Exponents, Polynomials, and Polynomial Functions

Radical Expressions and Equations

Quadratic Equations, Functions, and Inequalities

Exponential and Logarithmic Functions

Conic Sections; More Graphing

Miscellaneous Topics

Filters

a. What numbers cannot be solutions of the equation $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$ ? b. Solve: $\frac { x } { x - 5 } + \frac { 10 } { x } = 3$

Perform the operations and simplify the result when possible. $\frac { 5 x } { x ^ { 2 } + 11 x + 18 } + \frac { 5 x } { x ^ { 2 } - 4 }$

Multiply, then simplify if possible. $\frac { x + 3 } { 7 } \cdot \frac { x } { x + 3 }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 9 x + 18 \right) \div ( x - 3 )$

Add, and then simplify, if possible. $\frac { 7 } { x + 2 } + \frac { 8 } { x - 4 }$

Perform the division. $\frac { 6 a ^ { 3 } + a ^ { 2 } - 5 a + 7 } { a + 1 }$

Use synthetic division to perform the division. $\frac { 3 - 3 x ^ { 2 } + 2 x } { x - 4 }$

Perform the operations and simplify the result when possible. $\frac { x + 4 } { x + 5 } - \frac { x - 5 } { x + 7 }$

Find 3⁵ by using synthetic division to evaluate the polynomial $f ( x ) = x ^ { 5 } \text { at } x = 3$

Use synthetic division to perform the division. $\frac { 1 - 3 x ^ { 2 } + 2 x } { x - 4 }$

Divide using long division: $\frac { 2 x ^ { 3 } + 5 x ^ { 2 } - 1 } { x ^ { 2 } - 4 x + 2 }$

Divide $20 x ^ { 2 } - 7 x - 2$ by $4 x + 5$ .

Multiply, then simplify if possible. $\frac { ( x + 3 ) ^ { 2 } } { x + 8 } \cdot \frac { x + 8 } { x + 3 }$

If P ( x ) is a polynomial and if $P ( k ) = 0$ , then k is called a __________ of the polynomial.

Use the remainder theorem and synthetic division to find P ( k ). $P ( x ) = x ^ { 3 } - 3 x ^ { 2 } + x - 5 ; k = 2$ $P ( 2 ) =$ __________