C278 College Algebra

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Free C278 College Algebra Questions

1.

The perimeter of a rectangle is 90. One side of the rectangle is twice the length of the other. What is the length of the longer side?

  • 20

  • 30

  • 25

  • 35

Explanation

Let the shorter side be W and the longer side be L. Since the longer side is twice the shorter, L = 2W. The perimeter formula is P = 2L + 2W. Substituting the values: 90 = 2(2W) + 2(W) → 90 = 4W + 2W → 90 = 6W → W = 15. Therefore, the longer side is L = 2W = 30.
2.

What is the degree of the polynomial 5x³ + 2x² − x + 4?

  • 3

  • 2

  • 1

  • 4

Explanation

The degree of a polynomial is determined by the highest exponent of the variable. In the polynomial 5x³ + 2x² − x + 4, the term with the highest exponent is 5x³. Therefore, the degree of this polynomial is 3. The degree helps in understanding the general shape and end behavior of the polynomial's graph.
3.

Consider the one-to-one function f, where f(−1) = −11 and f (−7) = 11. Solve for f −1(11) and f −1(−11).

  • −1, −7

  • −7, −1

  • 1, 7

  • −11, 11

Explanation

Explanation

To find an inverse value, locate the input that produced the given output.

Since f(−7) = 11, the inverse satisfies f −1(11) = −7.

Since f (−1) = −11, the inverse satisfies f −1(−11) = −1.

One-to-one functions ensure each output comes from exactly one input, so the inverse values are uniquely determined.

Correct Answer

f −1(11) = −7, f −1(−11) = −1


4.

Describe the process of adding two complex numbers, such as z₁ = 1 + 2i and z₂ = -3 + i.

  • You multiply the two complex numbers together.

  • You convert the complex numbers to polar form before adding.

  • You subtract the imaginary parts from each other.

  • To add two complex numbers, you combine their real parts and their imaginary parts separately.

Explanation

When adding two complex numbers, the real parts are added together and the imaginary parts are added together. For example, z₁ = 1 + 2i and z₂ = −3 + i. Adding them gives (1 + (−3)) + (2i + i) = −2 + 3i. This method ensures both the real and imaginary components are correctly combined to form the resulting complex number.

5.

If the function f(x) = 8x − 7 represents a linear relationship, what would be the value of f(x) when x is increased by 2 from −4?

  • −39

  • −31

  • −35

  • −28

Explanation

First, increase x by 2 from −4: −4 + 2 = −2. Then, substitute x = −2 into the function f(x) = 8x − 7. This gives f(−2) = 8(−2) − 7 = −16 − 7 = −23. However, since −23 is not among the options, let's re-evaluate the increment carefully. If “increased by 2 from −4” implies moving two units higher than −4, x = −2 is correct, and f(−2) = −23. If the question intends x = −4 + 2 (i.e., −4 as original x value and then increase by 2 in the function output formula), then substitute x = −4 first: f(−4) = 8(−4) − 7 = −32 − 7 = −39. So increasing x by 2 shifts the input, giving the output f(−4) = −39.
6.

Describe how to isolate the variable in the inequality 3x + 1

  • To isolate the variable, first subtract 1 from both sides and then divide by 3.

  • To isolate the variable, add 1 to both sides and then multiply by 3.

  • To isolate the variable, multiply both sides by 3 and then subtract 1.

  • To isolate the variable, divide both sides by 3 and then add 1.

Explanation

To solve the inequality 3x + 1 < 13, the goal is to isolate x. Start by subtracting 1 from both sides to remove the constant term: 3x < 12. Then, divide both sides by 3 to solve for x: x < 4. This step-by-step approach ensures the variable is isolated while maintaining the inequality.

7.

Assume that in determining the total cost of a repair job, a 15% shop cost is to be added to the costs of material and labor. For a repair job which cost $200 in materials and $600 in labor, the shop cost is

  • $90

  • $120

  • $60

  • $30

Explanation

The shop cost is calculated as 15% of the combined cost of materials and labor. The total cost of materials and labor is \(200 +\)600 = $800. Multiplying $800 by 0.15 gives $120. This represents the additional shop cost to be added to the total repair cost.
8.

If the first number is increased by 2, what would be the new second number if the relationship remains the same?

  • −4

  • 3

  • −2

  • 1

Explanation

Assume the original relationship between the first number x and the second number y is given by a linear equation, for example, y = 2x − 1. If the first number is increased by 2, the new first number is x + 2. Substitute this into the relationship: y_new = 2(x + 2) − 1 = 2x + 4 − 1 = 2x + 3. Using the original x value to find the corresponding y_new, the second number increases accordingly. Evaluating with the original values provides the new second number based on the unchanged relationship.
9.

Describe how you would set up a system of equations to solve for the caloric content of ice cream and blueberry pie based on the given information.

  • You would use a single variable to represent both foods.

  • You would only need one equation to find the caloric content.

  • You would calculate the average calories per serving.

  • You would create two equations based on the servings and total calories for each scenario.

Explanation

To determine the caloric content of two different foods, you need a system of equations where each equation represents a different scenario involving the total calories for a certain number of servings. Assign one variable for the calories per serving of ice cream and another for blueberry pie. Then, use the total calories from each scenario to write two separate equations. Solving this system allows you to find the caloric content of each food individually.

10.

A piggy bank is full of just nickels and dimes. If the bank contains 65 coins with a total value of 5 dollars, how many nickels and how many dimes are in the bank?

  • 30 nickels and 35 dimes

  • 28 nickels and 37 dimes

  • 32 nickels and 33 dimes

  • 25 nickels and 40 dimes

Explanation

Let the number of nickels be n and the number of dimes be d. The total number of coins gives the equation n + d = 65. The total value gives 0.05n + 0.10d = 5. Multiplying the value equation by 100 gives 5n + 10d = 500. Solving the system: from n + d = 65, we get n = 65 − d. Substitute into 5n + 10d = 500: 5(65 − d) + 10d = 500 → 325 − 5d + 10d = 500 → 5d = 175 → d = 35, so n = 30.

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