Question 1

Find f ( c ) by evaluating directly and by using synthetic division and the remainder theorem. f ( x ) = x ² + 2 x + 4 and c = 9

Accepted Answer

The answer of Find f ( c ) by evaluating...

Question 2

Use similar triangles to help solve the problem. A man places a mirror on the ground and sees the reflection of the top of a flagpole as shown in the figure below. The two triangles in the figure are similar. Find the height h of the flagpole.  a = 13, b = 4, c = 34

Accepted Answer

The answer of Use similar triangles to help solve the...

Question 3

Use synthetic division to perform the division.

Accepted Answer

The answer of Use synthetic division to perform the division....

Question 4

If P ( y ) is a polynomial in y , then P ( r ) = 0 if and only if __________ is a factor of P ( y ).

Accepted Answer

If P(r) = 0, then this means that y - r is a factor of P(y) by the Remainder Theorem. This is because when we divide P(y) by y - r, the remainder will be 0, meaning that P(y) is evenly divisible by y - r. Therefore, (C) y - r is the correct answer.

Question 5

Use synthetic division to perform the division.

Accepted Answer

The answer of Use synthetic division to perform the division....

Question 6

A school board has determined that there should be 3 teachers for every 50 students. How many teachers are needed for an enrollment of 2,750 students?

Accepted Answer

The ratio of teachers to students is 3:50. For 2,750 students, divide 2,750 by 50 to find how many times the ratio is needed, then multiply by 3. $2,750 \div 50 = 55$, then $55 \times 3 = 165$ teachers.

Question 7

A model railroad engine is 2 inches long. If the scale is 168 feet to 1 foot, how long is the real engine?

Accepted Answer

The scale is 168 feet to 1 foot, meaning for every 1 foot in the model, the real engine is 168 feet. The model engine is 2 inches long, which is 1/6th of a foot (since there are 12 inches in a foot). Therefore, the real engine is 168 feet * 1/6 = 28 feet long.

Question 8

Use the factor theorem and tell whether the first expression is a factor of P ( x ). x + 1; P ( x ) = x ³ + 3 x ² - 3 x - 5 . (Hint: Write x + 1 as x - (-1).)

Accepted Answer

To use the factor theorem, we need to determine if P(-1) = 0. Let's substitute -1 for x in P(x): P(-1) = (-1)^3³ + 3(-1)^2² - 3(-1) - 5 = -³ + 3² + 2 We can see that P(-1) is not equal to 0. However, we can rewrite x + 1 as x - (-1), and then substitute -1 for x - (-1) in P(x): P(x - (-1)) = (x - (-1))^3³ + 3(x - (-1))^2² - 3(x - (-1)) - 5 = (x + 1)^3³ + 3(x + 1)^2² - 3(x + 1) - 5 = (x^3 + 3x^2 + 3x + 1)³ + (3x^2 + 6x + 3)² - 3x - 3 - 5 = x^3³ + 3x^2³ + 3x³ + ³ + 3x^2² + 6x² + 3² - 3x - 8 = x^3(³) + x^2(3³ + 3²) + x(3³ + 6² - 3) + (³ + 3² - 8) We can see that (x+1) is indeed a factor of P(x) because P(x-(-1)) is equal to (x+1) times another polynomial. Therefore, the answer is A.

Question 9

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Accepted Answer

Let x be the height of the tree.
We have the proportion: 
5/4 = x/24
Simplifying, we get:
x = 30
Therefore, the height of the tree is 30 ft.

Question 10

Perform the division:  by  .

Accepted Answer

The answer of Perform the division:  by  ....

Question 11

Use a calculator and synthetic division to perform the division.

Accepted Answer

The answer of Use a calculator and synthetic division to...

Question 12

Solve the proportion for the variable.  z = __________

Accepted Answer

The answer of Solve the proportion for the variable. ...

Question 13

Express the sentence as a formula. Assume that  is a constant of variation. The distance  that a body falls varies directly with the square of the time  .

Accepted Answer

The answer of Express the sentence as a formula. Assume...

Question 14

Assume that all variables represent positive numbers. Assume that  varies directly with  . If  when  , find  when  .

Accepted Answer

If 11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 when 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11, then we can write:

11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 = k * 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11

where k is a constant of proportionality.

Also, we know that 11ec1e8e_2c09_29a6_be9b_dfe174236238_TBX8967_11 varies directly with 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11. So we can write:

11ec1e8e_2c09_29a6_be9b_dfe174236238_TBX8967_11 = m * 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11

where m is a constant of proportionality.

We need to find 11ec1e8e_2c09_29aa_be9b_7950c7064e55_TBX8967_11 when 11ec1e8e_2c09_50bb_be9b_c1303cde95c0_TBX8967_11. If we substitute the given values in the above equations, we get:

11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 = k * 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11

11ec1e8e_2c09_50bb_be9b_c1303cde95c0_TBX8967_11 = m * 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11

We can solve for k and m:

k = 11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 / 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11

m = 11ec1e8e_2c09_50bb_be9b_c1303cde95c0_TBX8967_11 / 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11

Now we can substitute these values to find 11ec1e8e_2c09_29aa_be9b_7950c7064e55_TBX8967_11:

11ec1e8e_2c09_29aa_be9b_7950c7064e55_TBX8967_11 = k * m * 11ec1e8e_2c09_29a6_be9b_dfe174236238_TBX8967_11

= (11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 / 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11) * (11ec1e8e_2c09_50bb_be9b_c1303cde95c0_TBX8967_11 / 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11) * 11ec1e8e_2c09_29a6_be9b_dfe174236238_TBX8967_11

Simplifying this expression gives:

11ec1e8e_2c09_29aa_be9b_7950c7064e55_TBX8967_11 = (11ec1e8e_2c09_29a8_be9b_d1273849166a_TBX8967_11 * 11ec1e8e_2c09_50bb_be9b_c1303cde95c0_TBX8967_11 / 11ec1e8e_2c09_29a9_be9b_e1306c6f56e5_TBX8967_11) * (11ec1e8e_2c09_29a6_be9b_dfe174236238_TBX8967_11 / 11ec1e8e_2c09_29a7_be9b_514302ce0e3f_TBX8967_11)

Therefore, the correct answer is D) 11ec1e8e_2c09_50bf_be9b_0f76536c5018_TBX8967_11.

Question 15

Use the factor theorem and tell whether the first expression is a factor of P ( x ). x ; P ( x ) = 7 x ³ - 7 x ² - 8 x . Hint : Write x as x - 0.

Accepted Answer

Using the factor theorem, if (x-a) is a factor of P(x), then P(a) = 0. 
In this case, we can write x as (x-0), so we need to check if P(0) = 0. Plugging in 0, we get:
P(0) = 7(0)^3 - 7(0)^2 - 8(0) = 0
Since P(0) = 0, we can conclude that (x-0) or simply x is a factor of P(x). Therefore, the answer is B.

Question 16

Let P ( x ) = 4 x ³- 3 x ²+ 2 x - 5. Evaluate P ( x ) by substituting the given value of x into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. P (-4)

Accepted Answer

The answer of Let P ( x ) = 4...

Question 17

Solve the proportion for the variable.  y = __________

Accepted Answer

The answer of Solve the proportion for the variable. ...

Question 18

Solve for the variable in the proportion.

Accepted Answer

The answer of Solve for the variable in the proportion....

Question 19

Use similar triangles to help solve the problem. An airplane ascends 160 feet as it flies a horizontal distance of 1,000 feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? ( Hint: 5,280 feet = 1 mile.) Round your answer to the nearest integer.

Accepted Answer

By using similar triangles, we can set up the following proportion:

$\frac{160}{1000} = \frac{x}{5280}$

where x is the altitude gain over 1 mile. Solving for x:

$x = \frac{160*5280}{1000} \approx \boxed{845 	ext{ ft}}$

Question 20

A recipe for spaghetti sauce requires 6 bottles of ketchup to make 8 gallons of sauce. How many bottles of ketchup are needed to make 56 gallons of sauce?

Accepted Answer

The answer of A recipe for spaghetti sauce requires 6...

Quiz 6: Rational Expressions

Find f ( c ) by evaluating directly and by using synthetic division and the remainder theorem. f ( x ) = x ² + 2 x + 4 and c = 9

Use similar triangles to help solve the problem. A man places a mirror on the ground and sees the reflection of the top of a flagpole as shown in the figure below. The two triangles in the figure are similar. Find the height h of the flagpole. a = 13, b = 4, c = 34

Use synthetic division to perform the division.

If P ( y ) is a polynomial in y , then P ( r ) = 0 if and only if __________ is a factor of P ( y ) .

Use synthetic division to perform the division.

A school board has determined that there should be 3 teachers for every 50 students. How many teachers are needed for an enrollment of 2,750 students?

A model railroad engine is 2 inches long. If the scale is 168 feet to 1 foot, how long is the real engine?

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x + 1; P ( x ) = x ³ + 3 x ² - 3 x - 5 . (Hint: Write x + 1 as x - (-1) .)

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Perform the division: by .

Use a calculator and synthetic division to perform the division.

Solve the proportion for the variable. z = __________

Express the sentence as a formula. Assume that is a constant of variation. The distance that a body falls varies directly with the square of the time .

Assume that all variables represent positive numbers. Assume that varies directly with . If when , find when .

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x ; P ( x ) = 7 x ³ - 7 x ² - 8 x . Hint : Write x as x - 0.

Let P ( x ) = 4 x ³- 3 x ²+ 2 x - 5. Evaluate P ( x ) by substituting the given value of x into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. P (-4)

Solve the proportion for the variable. y = __________

Solve for the variable in the proportion.

Use similar triangles to help solve the problem. An airplane ascends 160 feet as it flies a horizontal distance of 1,000 feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? ( Hint: 5,280 feet = 1 mile.) Round your answer to the nearest integer.

A recipe for spaghetti sauce requires 6 bottles of ketchup to make 8 gallons of sauce. How many bottles of ketchup are needed to make 56 gallons of sauce?

Quiz 6: Rational Expressions

Find f ( c ) by evaluating directly and by using synthetic division and the remainder theorem. f ( x ) = x 2 + 2 x + 4 and c = 9

Use similar triangles to help solve the problem. A man places a mirror on the ground and sees the reflection of the top of a flagpole as shown in the figure below. The two triangles in the figure are similar. Find the height h of the flagpole. a = 13, b = 4, c = 34

Use synthetic division to perform the division.

If P ( y ) is a polynomial in y , then P ( r ) = 0 if and only if __________ is a factor of P ( y ) .

Use synthetic division to perform the division.

A school board has determined that there should be 3 teachers for every 50 students. How many teachers are needed for an enrollment of 2,750 students?

A model railroad engine is 2 inches long. If the scale is 168 feet to 1 foot, how long is the real engine?

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x + 1; P ( x ) = x 3 + 3 x 2 - 3 x - 5 . (Hint: Write x + 1 as x - (-1) .)

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Perform the division: by .

Use a calculator and synthetic division to perform the division.

Solve the proportion for the variable. z = __________

Express the sentence as a formula. Assume that is a constant of variation. The distance that a body falls varies directly with the square of the time .

Assume that all variables represent positive numbers. Assume that varies directly with . If when , find when .

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x ; P ( x ) = 7 x 3 - 7 x 2 - 8 x . Hint : Write x as x - 0.

Let P ( x ) = 4 x 3 - 3 x 2 + 2 x - 5. Evaluate P ( x ) by substituting the given value of x into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. P (-4)

Solve the proportion for the variable. y = __________

Solve for the variable in the proportion.

Use similar triangles to help solve the problem. An airplane ascends 160 feet as it flies a horizontal distance of 1,000 feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? ( Hint: 5,280 feet = 1 mile.) Round your answer to the nearest integer.

A recipe for spaghetti sauce requires 6 bottles of ketchup to make 8 gallons of sauce. How many bottles of ketchup are needed to make 56 gallons of sauce?

Find f ( c ) by evaluating directly and by using synthetic division and the remainder theorem. f ( x ) = x ² + 2 x + 4 and c = 9

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x + 1; P ( x ) = x ³ + 3 x ² - 3 x - 5 . (Hint: Write x + 1 as x - (-1) .)

Use the factor theorem and tell whether the first expression is a factor of P ( x ) . x ; P ( x ) = 7 x ³ - 7 x ² - 8 x . Hint : Write x as x - 0.

Let P ( x ) = 4 x ³- 3 x ²+ 2 x - 5. Evaluate P ( x ) by substituting the given value of x into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. P (-4)