C278 College Algebra

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Free C278 College Algebra Questions

1.

If you have the polynomials (3x⁴ + 7x² − 2) and (5x⁴ − 4x² + 9), what would be the result of (3x⁴ + 7x² − 2) − (5x⁴ − 4x² + 9) in standard form?

  • 2x⁴ + 11x² + 11

  • −2x⁴ + 11x² − 11

  • 2x⁴ − 11x² + 11

  • −2x⁴ + 3x² + 7

Explanation

To subtract the polynomials, distribute the negative sign to each term of the second polynomial: (3x⁴ + 7x² − 2) − 5x⁴ + 4x² − 9. Then combine like terms: 3x⁴ − 5x⁴ = −2x⁴, 7x² + 4x² = 11x², and −2 − 9 = −11. Therefore, the resulting polynomial in standard form is −2x⁴ + 11x² − 11.
2.

Given the functions m(x) = -8x + 14 and n(x) = -4x + 5, find m(n(-3))

  • 17

  • 3

  • −147

  • −122

  • 136

Explanation

Explanation

First compute n(-3). Substitute into n(x) = -4x + 5:

n(-3) = -4(-3) + 5 = 12 + 5 = 17.

Next, substitute this result into m(x).

m(17) = -8(17) + 14.

Compute: -8 . 7 = -136.

Then add 14: -136 + 14 = -122.

Thus, m(n(-3) = -122.

Correct Answer

−122


3.

Describe the significance of the slope in the context of a linear equation.

  • The slope shows the maximum value of the function.

  • The slope determines the degree of the polynomial.

  • The slope represents the rate of change of the dependent variable with respect to the independent variable.

  • The slope indicates the y-intercept of the line.

Explanation

In a linear equation of the form y = mx + b, the slope, m, represents how much the dependent variable (y) changes for a unit change in the independent variable (x). It indicates the steepness and direction of the line. A positive slope means the line rises, while a negative slope means it falls. Understanding the slope is crucial for interpreting relationships between variables in real-world contexts.

4.

What is the first step to solve the linear inequality 3x + 1

  • Add 1 to both sides.

  • Subtract 1 from both sides.

  • Divide both sides by 3.

  • Multiply both sides by 3.

Explanation

To solve the inequality 3x + 1 < 13, the first step is to isolate the variable term by removing the constant on the left side. This is done by subtracting 1 from both sides, giving 3x < 12. This step simplifies the inequality and prepares it for the next step of dividing by 3 to solve for x.

5.

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.
6.

If the equation 2x/5 = x/3 + 5 represents a real-world scenario where x is the number of hours worked, how would you interpret the solution you find for x?

  • The solution represents the number of hours worked to achieve a certain income level.

  • The solution is irrelevant to real-world applications.

  • The solution shows the maximum hours one can work in a week.

  • The solution indicates the total income earned.

Explanation

In this equation, x represents the number of hours worked, and the equation models a relationship between hours and another quantity, such as income. Solving for x provides the specific number of hours required to satisfy the given condition (for example, to earn a certain total). Therefore, the solution directly corresponds to a meaningful, real-world value—the number of hours worked to reach the target.

7.

If the quadratic equation x² − 5x + 4 = 0 represents the area of a rectangle where the length is x and the width is (x − 4), what is the width when the area is zero?

  • 4

  • 1

  • 0

  • 5

Explanation

The area of a rectangle is given by length × width. Here, the length is x and the width is x − 4, so the area is A = x(x − 4). Setting the area to zero gives the equation x(x − 4) = 0. Solving this yields x = 0 or x = 4. Since the width is x − 4, substitute these values: for x = 0, width = 0 − 4 = −4 (not physically meaningful), and for x = 4, width = 4 − 4 = 0. Therefore, the width is zero when the area is zero.
8.

Describe the significance of the discriminant in the quadratic formula.

  • The discriminant indicates the slope of the parabola.

  • The discriminant determines the nature of the roots of the quadratic equation.

  • The discriminant provides the vertex of the parabola.

  • The discriminant shows the maximum value of the quadratic function.

Explanation

The discriminant in the quadratic formula is the part under the square root: b² − 4ac. It determines the nature of the roots of a quadratic equation. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one repeated real root; and if it is negative, the roots are complex or imaginary. Thus, the discriminant gives critical information about whether solutions exist and what type they are.

9.

What formula is used to calculate the perimeter of a rectangle?

  • P = 2L + 2W

  • P = L * W

  • P = 2(L + W)

  • P = L + W

Explanation

The perimeter of a rectangle is the total distance around the shape. A rectangle has two lengths (L) and two widths (W), so the perimeter can be calculated as the sum of all sides: P = L + W + L + W = 2L + 2W. This is equivalent to the formula P = 2(L + W), which is the standard formula used for calculating the perimeter efficiently.
10.

What is the total cost of the items purchased by the first customer?

  • $20.00

  • $15.50

  • $11.25

  • $16.75

Explanation

To find the total cost, add the prices of all items purchased by the first customer. For example, if the first customer bought items with costs that sum up to $11.25, then that amount represents the total expenditure. Calculating the sum ensures the correct total for the customer's purchase.

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