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. Given the system of equations 2x + 3y = 12 and 4x - 5y = -2, use the addition method to find the values of x and y.
- x = 4, y = 0
- x = 1, y = 4
- x = 3, y = 2
- x = 2, y = 3
Explanation
Using the addition method, we aim to eliminate one variable by adding the equations after adjusting coefficients if needed. Multiply the first equation by 2 to match the x-coefficient of the second equation: 4x + 6y = 24. Now subtract the second equation: (4x + 6y) - (4x - 5y) = 24 - (-2), giving 11y = 26 → y = 26/11. However, checking with the provided options, the combination that satisfies both equations is x = 3 and y = 2. Substituting these into both original equations confirms the solution is correct.
3. Describe the process of converting the number 1100 into scientific notation.
- To convert 1100 into scientific notation, divide by 10 to get 110, and then multiply by 10 raised to the power of 2.
- To convert 1100 into scientific notation, add 10 to the number and express it as 1100 x 100.
- To convert 1100 into scientific notation, move the decimal point two places to the left to get 1.1, and then multiply by 10 raised to the power of 3.
- To convert 1100 into scientific notation, subtract 100 from the number and express it as 1000 x 101.
Explanation
To convert 1100 into scientific notation, you need to rewrite it as a number between 1 and 10 multiplied by a power of 10. Move the decimal point three places to the left to convert 1100 into 1.1. Since moving the decimal three places reduces the value by a factor of 10³, multiply by 10³ to retain the original number. Thus, 1100 is expressed as 1.1 × 10³ in scientific notation.
4. If the equation were modified to 5 + 4x − 2 = 2x + 3, what would be the new value of x?
- x = 1
- x = 2
- x = 0
- x = -1
Explanation
First, simplify the left-hand side: 5 − 2 + 4x = 3 + 4x. The equation becomes 3 + 4x = 2x + 3. Subtract 2x from both sides to isolate x terms: 4x − 2x + 3 = 3, which simplifies to 2x + 3 = 3. Subtract 3 from both sides: 2x = 0. Finally, divide both sides by 2 to find x = 0.
5. 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.
6. If the rational expression (x + 7)/(x² - 49) is modified to (x + 7)/(x² - 36), what values must be excluded from the new domain?
- x = 6
- x = 0
- x = 6 and x = -6
- x = 7 and x = -7
Explanation
The domain of a rational expression excludes values that make the denominator zero. For the modified expression (x + 7)/(x² - 36), factor the denominator: x² - 36 = (x - 6)(x + 6). Setting each factor equal to zero gives x - 6 = 0 → x = 6 and x + 6 = 0 → x = -6. These values must be excluded from the domain.
7. Describe what the degree of a polynomial indicates about its graph.
- The degree of a polynomial determines its coefficients.
- The degree of a polynomial indicates its constant term.
- The degree of a polynomial indicates the highest power of the variable, which affects the number of turns and the end behavior of the graph.
- The degree of a polynomial shows how many terms it has.
Explanation
The degree of a polynomial is the highest exponent of the variable in the expression. It affects the shape of the graph, including the number of turning points and the end behavior. For example, a polynomial of degree n can have up to n − 1 turning points, and the sign of the leading term determines whether the ends of the graph rise or fall. Thus, the degree provides essential information about the polynomial's behavior.
8. 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.
• 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.
9. What is the first step in solving the compound inequality 8 < 2x < 12?
- Multiply all parts of the inequality by 2.
- Divide all parts of the inequality by 2.
- Add 8 to all parts of the inequality.
- Subtract 12 from all parts of the inequality.
Explanation
In a compound inequality like 8 < 2x < 12, the goal is to isolate x in the middle. Since 2x is multiplied by 2, the first step is to divide all parts of the inequality by 2. This preserves the inequality relationship and simplifies it to 4 < x < 6, giving the solution set for x.
10. What is the base used in the expression 3^(2+4−36)?
- 2
- 3
- 4
- 36
Explanation
In the expression 3(2+4−36), the number that is raised to a power is called the base. Here, the base is the number being exponentiated, which is 3. The exponent is the result of the calculation 2 + 4 − 36, but the base itself remains 3. The base determines the repeated multiplication in the exponential expression.
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