C278 College Algebra

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Free C278 College Algebra Questions

1. Describe the relationship between the president's salary and the department head's salary based on the information provided.

The president earns the same salary as the department head.
The president earns three times the salary of the department head.
The president earns half the salary of the department head.
The president earns twice the salary of the department head.

Explanation

Based on typical salary information provided, the president’s salary is generally set at a specific multiple of other administrative roles, such as department heads. If the information indicates that the president’s salary is $400,000 while the department head earns $200,000, the president earns exactly twice the salary of the department head. This shows a direct proportional relationship between the two salaries.

2. If you have the polynomial expression 4x² − 3x + 7 and you add the polynomial 2x³ + x, what will be the degree of the resulting polynomial?

Explanation

The degree of a polynomial is determined by the term with the highest exponent. Adding 4x² − 3x + 7 and 2x³ + x gives 2x³ + 4x² − 2x + 7. The term with the highest exponent is 2x³, which has an exponent of 3. Therefore, the degree of the resulting polynomial is 3.

3. Describe the process of solving the linear inequality -2x + 9 > x + 3 in your own words.

To solve the inequality, you can ignore the inequality sign.
To solve the inequality, first isolate x by moving all x terms to one side and constant terms to the other side.
To solve the inequality, you need to factor the expression completely.
To solve the inequality, you should only focus on the constant terms.

Explanation

When solving a linear inequality such as −2x + 9 > x + 3, the goal is to isolate x. Start by adding 2x to both sides to eliminate the negative x term on the left, giving 9 > 3x + 3. Then, subtract 3 from both sides to get 6 > 3x. Finally, divide both sides by 3 to find 2 > x, or equivalently, x < 2. The key steps involve moving all x terms to one side and all constants to the other while preserving the direction of the inequality.

4. The sum of two polynomials is 8d⁵ − 3c³d² + 5c²d³ − 4cd⁴ + 9. If one addend is 2d⁵ − c³d² + 8cd⁴ + 1, what is the other addend?

6d5 − 2c3d2 + 5c2d3 − 12cd4 + 8
6d5 − 4c3d2 + 5c2d3 − 12cd4 + 8
6d5 − 4c3d2 + 3c2d3 − 4cd4 + 8
6d5 − 2c3d2 − 3c2d3 − 4cd4 + 8

Explanation

To find the other addend, subtract the known addend from the sum of the polynomials. Subtract term by term:
• For d⁵ terms: 8d⁵ − 2d⁵ = 6d⁵
• For c³d² terms: −3c³d² − (−c³d²) = −3c³d² + c³d² = −2c³d²
• For c²d³ terms: 5c²d³ − 0 = 5c²d³
• For cd⁴ terms: −4cd⁴ − 8cd⁴ = −12cd⁴
• For constants: 9 − 1 = 8
Combining all results gives 6d⁵ − 2c³d² + 5c²d³ − 12cd⁴ + 8.

5. For the equation 4X + 2 = 10, X equals?

2
12
3
-2

Explanation

To solve 4X + 2 = 10, start by isolating the variable term. Subtract 2 from both sides: 4X = 8. Then, divide both sides by 4 to solve for X: X = 8 ÷ 4 = 2. This gives the value of X that satisfies the equation.

6. If a different boat sold for $15,000 after a 20% price reduction, what was the original price of that boat?

$12,000
$18,750
$15,000
$20,000

Explanation

Let the original price be P. After a 20% reduction, the selling price is 80% of the original price: 0.8 × P = 15,000. Solving for P gives P = 15,000 ÷ 0.8 = 18,750. Thus, the original price of the boat was $18,750.

7. What is the minimum value of the function f(x) = 3x² + 3x − 5?

−5.75
−5.00
−8.0
−12.0

Explanation

For a quadratic function f(x) = ax² + bx + c with a > 0, the parabola opens upward, so the minimum value occurs at the vertex. The x-coordinate of the vertex is given by x = −b/(2a). Here, a = 3 and b = 3, so x = −3/(2×3) = −0.5. Substitute x = −0.5 into the function: f(−0.5) = 3(−0.5)² + 3(−0.5) − 5 = 3(0.25) − 1.5 − 5 = 0.75 − 1.5 − 5 = −5.75. Therefore, the minimum value of the function is −5.75.

8. What are the factors of the quadratic equation x² − 5x + 4?

(x + 2)(x − 2)
(x − 2)(x − 2)
(x + 1)(x + 4)
(x − 1)(x − 4)

Explanation

To factor x² − 5x + 4, find two numbers that multiply to 4 (the constant term) and add up to −5 (the coefficient of x). These numbers are −4 and −1. Therefore, the quadratic can be factored as (x − 4)(x − 1). Expanding this gives x² − x − 4x + 4, which simplifies back to x² − 5x + 4, confirming the correct factorization.

Consider the table of values below and find the linear function f(x) that corresponds to the table.

4x − 3
4x + 3
3x − 4
2x − 3

Explanation

Explanation

The outputs increase by 4 each step in x, so the slope is m=4. For a linear form f(x) = mx + b, plug in x = 0 ⇒ f(0)= −3 to get b = −3. Therefore, the function matching the table is f(x) = 4x − 3.

Correct Answer

4x − 3

10. Describe the process of evaluating the function f(x) = 3x² − 4x + 2 at x = x − 1.

Add 1 to the function and evaluate at x.
Factor the function before substituting x − 1.
Substitute x − 1 into the function and simplify the expression.
Differentiate the function and evaluate at x − 1.

Explanation

To evaluate the function f(x) = 3x² − 4x + 2 at x = x − 1, replace every occurrence of x in the function with (x − 1). This gives f(x − 1) = 3(x − 1)² − 4(x − 1) + 2. Then, expand and simplify the expression to find the value of the function at this input. This substitution method is standard when evaluating a function at a shifted input.

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