Q1. Which of the following is a factor of the polynomial x^3 - 3x^2 - 4x + 12?
Solution:
Using synthetic division or the factor theorem, we can test x = 2. Substituting gives us 2^3 - 3(2^2) - 4(2) + 12 = 0, confirming that x - 2 is a factor.
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Q2. What is the solution set for the equation 3x + 2 = 11?
Solution:
To solve for x, subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3.
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Q3. What is the solution set for the inequality x^2 - 5x + 6 > 0?
Solution:
Factoring gives (x - 2)(x - 3) > 0. The roots are x = 2 and x = 3. The solution set is x < 2 or x > 3.
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Q4. If f(x) = x^2 - 6x + 8, what are the roots of f(x)?
Solution:
Factoring gives f(x) = (x - 2)(x - 4). Setting each factor to zero gives the roots x = 2 and x = 4.
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Q5. What is the value of x in the equation 4x - 12 = 0?
Solution:
Add 12 to both sides: 4x = 12. Then divide by 4: x = 3.
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Q6. What is the value of x in the equation 3x + 2 = 11?
Solution:
Subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3.
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Q7. Which of the following represents the factored form of x^2 + 7x + 10?
Solution:
To factor x^2 + 7x + 10, we look for two numbers that multiply to 10 and add to 7. The numbers 5 and 2 work, so the factored form is (x + 5)(x + 2).
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Q8. Which of the following is a root of the polynomial x^2 + 3x - 10?
Solution:
Factoring gives (x + 5)(x - 2) = 0. The roots are x = -5 and x = 2, so 2 is a root.
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Q9. Which of the following is a factor of the polynomial x^3 - 4x^2 + 4x?
Solution:
We can factor out x from the polynomial: x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x - 2)^2. Thus, x is a factor.
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Q10. What is the solution to the inequality x^2 - 5x + 6 > 0?
Solution:
Factoring gives (x - 2)(x - 3) > 0. The roots are x = 2 and x = 3. The solution set is x < 2 or x > 3.
