Polynomials - Roots and Factor Theorem

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Q1. What are the roots of the polynomial x^2 - 4?

Solution:

x^2 - 4 can be factored as (x - 2)(x + 2) = 0. Therefore, the roots are -2 and 2.

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Q2. What is the factored form of x^2 - 5x + 6?

Solution:

The polynomial factors to (x - 2)(x - 3) since the roots are 2 and 3.

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Q3. Which of the following is a root of the polynomial x^3 - 6x^2 + 11x - 6?

Solution:

Using the Rational Root Theorem, we can test x = 1, 2, and 3. Testing x = 3 gives 3^3 - 6(3^2) + 11(3) - 6 = 0, so x = 3 is a root.

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Q4. If x^2 - 4x + 4 = 0, what is the repeated root?

Solution:

Factoring gives (x - 2)(x - 2) = 0, so the repeated root is x = 2.

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Q5. Which polynomial has a root at x = -1?

Solution:

The polynomial x^2 + 2x + 1 can be factored as (x + 1)(x + 1), indicating that -1 is a root.

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Q6. If f(x) = x^2 - 9, what are the roots of the polynomial?

Solution:

We can factor the polynomial as f(x) = (x - 3)(x + 3). Setting each factor to zero gives us the roots x = -3 and x = 3.

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Q7. Which of the following represents the roots of the equation 2x^2 - 8 = 0?

Solution:

First, simplify the equation: 2x^2 = 8, so x^2 = 4. The roots are x = ±2.

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Q8. If x^2 - 6x + 9 = 0, what is the repeated root?

Solution:

The polynomial can be factored as (x - 3)(x - 3) = 0, giving a repeated root of x = 3.

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Q9. What is the value of k if x^2 - kx + 12 has roots 3 and 4?

Solution:

Using the sum of roots, 3 + 4 = k, we find k = 7.

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Q10. Which of the following is a solution to the equation x^2 - 7x + 10 = 0?

Solution:

Factoring gives (x - 5)(x - 2) = 0, so the solutions are x = 5 and x = 2.