Polynomials - Roots and Factor Theorem

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Q1. Which inequality represents the solution to the equation 4x - 8 = 0?

Solution:

To solve the equation, add 8 to both sides: 4x = 8. Then divide by 4: x = 2. The solution is x = 2.

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Q2. If f(x) = x^2 - 6x + 9, what is f(3)?

Solution:

Substituting x = 3 into the function: f(3) = 3^2 - 6(3) + 9 = 9 - 18 + 9 = 0.

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Q3. What is the solution to the equation 3x + 2 = 11?

Solution:

Subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3.

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Q4. What is the factored form of the polynomial 3x^2 - 12?

Solution:

First, factor out 3: 3(x^2 - 4). Then, factor the difference of squares: 3(x - 2)(x + 2).

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Q5. Which of the following is the solution set for the inequality x + 3 > 2?

Solution:

To solve the inequality, subtract 3 from both sides: x > -1.

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Q6. What are the roots of the polynomial x^2 + 4x + 4?

Solution:

The polynomial can be factored as (x + 2)(x + 2) or (x + 2)^2. Thus, the roots are x = -2 and x = -2.

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Q7. Which of the following represents the solution set for the inequality x + 2 > 3?

Solution:

Subtracting 2 from both sides gives x > 1.

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Q8. What is the discriminant of the quadratic equation x^2 - 6x + 9?

Solution:

The discriminant is given by b^2 - 4ac. Here, a = 1, b = -6, and c = 9. So, the discriminant is (-6)^2 - 4(1)(9) = 36 - 36 = 0.