C278 College Algebra
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Free C278 College Algebra Questions
1. If the equation changes to 3(2-x) = 81, what is the value of x?
- 2
- 3
- 4
- 1
Explanation
To solve 3(2−x) = 81, first express 81 as a power of 3. Since 81 = 3⁴, the equation becomes 3(2−x) = 3⁴. Because the bases are the same, the exponents must be equal: 2 − x = 4. Solving for x gives x = 2 − 4 = −2. However, since none of the provided options include −2, let’s recheck the direction of subtraction: 3(2−x) = 3⁴ implies 2 − x = 4, which still gives x = −2. That indicates there might be a typographical oversight in the options, but if instead it were 3(x−2) = 81, then x − 2 = 4, so x = 6. Given the context and typical intent of such a problem, the correct step based on 3(2−x) = 81 still leads mathematically to x = −2.
2. Describe how to determine the degree of a polynomial expression.
- The degree of a polynomial is always equal to the number of variables present.
- The degree of a polynomial is determined by identifying the highest power of the variable in the expression.
- The degree of a polynomial is the sum of the coefficients of the terms.
- The degree of a polynomial is the number of terms in the expression.
Explanation
The degree of a polynomial is the highest exponent of the variable in the polynomial. For a single-variable polynomial, it is simply the largest power of x. For multivariable polynomials, the degree of a term is the sum of the exponents of all variables in that term, and the degree of the polynomial is the highest such sum among all terms. Identifying the term with the largest power or sum of powers allows you to determine the degree of the polynomial accurately.
3. Describe the relationship between the first and second numbers based on the equation provided in the problem.
- The first number is twice the second number.
- The first number is always greater than the second number.
- The sum of the two numbers is always positive.
- Twice the second number equals 4 times the first number minus 2.
Explanation
The relationship between two numbers is determined by the equation that links them. If the equation shows that one number can be expressed as a multiple of the other, then that defines their relationship. For example, if the first number equals twice the second number, then for any value of the second number, the first number will always be exactly twice as large. This shows a direct proportional relationship between the first and second numbers.
4. Describe how the substitution method works in solving systems of equations.
- The substitution method involves multiplying both equations by a common factor.
- The substitution method involves solving one equation for a variable and substituting that expression into the other equation.
- The substitution method is only applicable to linear equations.
- The substitution method requires adding both equations to eliminate a variable.
Explanation
The substitution method is a technique for solving systems of equations by first solving one of the equations for one variable in terms of the other. This expression is then substituted into the second equation, effectively reducing the system to a single equation with one variable. Solving this equation yields the value of one variable, which can then be substituted back to find the other variable. This method works for both linear and some nonlinear systems, but it is most commonly applied to linear systems.
5. F(x) = 3x² − 6x − 45 factors most completely as:
- (3x+9)(x-5)
- 3(x+3)(x-5)
- −3(3x+3)(x-5)
- (x+3)(2x-15)
Explanation
To factor 3x² − 6x − 45, first factor out the greatest common factor, which is 3: 3(x² − 2x − 15). Next, factor the quadratic x² − 2x − 15. Find two numbers that multiply to −15 and add to −2, which are −5 and 3. Thus, x² − 2x − 15 factors as (x − 5)(x + 3). Combining this with the GCF gives 3(x + 3)(x − 5).
6. Describe how the vertical line test can be applied to the equation y = -x - 7 to confirm if it is a function.
- If the graph is a straight line, it cannot be a function.
- If any vertical line intersects the graph at most once, then y is a function of x.
- If the graph has more than one y-value for a single x-value, it is a function.
- If the graph is curved, it is not a function.
Explanation
The vertical line test determines if a graph represents a function by checking whether any vertical line intersects the graph more than once. For the linear equation y = -x - 7, every vertical line intersects the line at exactly one point. This confirms that each x-value has a unique y-value, satisfying the definition of a function.
7. If you were to factor the polynomial 98x² - 72, what would be the first step in the factoring process?
- Rewrite the polynomial as a sum of squares.
- Set the polynomial equal to zero and solve for x.
- Factor out the greatest common factor, which is 2.
- Use the quadratic formula to find the roots.
Explanation
Factoring a polynomial often begins by identifying and factoring out the greatest common factor (GCF) of all terms. In the polynomial 98x² - 72, both coefficients 98 and 72 are divisible by 2. Factoring out 2 simplifies the polynomial, making it easier to apply further factoring methods, such as factoring a difference of squares if applicable. This first step ensures the polynomial is in its simplest form before attempting more complex factoring.
8. Describe how you would set up a system of equations to solve for the cost of hot dogs and potato chips based on the purchases made by the two customers.
- You would only need one equation since both customers bought the same items.
- You would create two equations: one for each customer's purchase, using variables for the cost of hot dogs and potato chips.
- You would use the total cost to find the average price of the items.
- You would set up a single equation with the total cost of both customers combined.
Explanation
To determine the cost of hot dogs and potato chips, assign variables to each item, for example, let x represent the cost of a hot dog and y represent the cost of a bag of potato chips. Each customer’s purchase gives an equation based on the total amount they spent. With two different purchases, you can create a system of two equations with two variables. Solving this system simultaneously will provide the values of x and y. This approach is standard for solving problems involving multiple unknowns with separate total amounts.
9. What is the vertical line test used for in graphing equations?
- To identify the x-intercepts of a graph.
- To determine if a graph represents a function.
- To find the slope of a line.
- To calculate the area under a curve.
Explanation
The vertical line test is a visual method used to determine whether a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function because a single x-value corresponds to multiple y-values. If every vertical line intersects the graph at most once, the graph passes the test and represents a function.
10. What is the general effect of multiplying two complex numbers?
- It rotates and scales the complex number in the complex plane
- It results in a purely imaginary number
- It always reduces the combined phase of the complex numbers
- It always increases the magnitude of the resulting complex number
Explanation
When two complex numbers are multiplied, their magnitudes are multiplied and their arguments (angles) are added. This means that the product experiences both a scaling (change in magnitude) and a rotation (change in angle) in the complex plane. The resulting effect is not necessarily an increase or decrease in size alone—it depends on the magnitudes of the two numbers—but the combination of rotation and scaling is consistent. This geometric interpretation makes multiplication of complex numbers unique.
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