Factorization Techniques

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Q1. Factor the polynomial x^3 - 3x^2 - 4x.

Solution:

First, factor out the common term x: x(x^2 - 3x - 4). Now, factor the quadratic x^2 - 3x - 4 to get x(x - 4)(x + 1).

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Q2. Factor the expression 2x^2 - 8.

Solution:

To factor 2x^2 - 8, first factor out 2: 2(x^2 - 4). Then, recognize x^2 - 4 as a difference of squares: 2(x - 2)(x + 2).

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Q3. Solve the equation 3x + 7 = 16.

Solution:

To solve 3x + 7 = 16, subtract 7 from both sides: 3x = 9. Then divide by 3: x = 3.

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Q4. Solve the equation 2x^2 - 8 = 0.

Solution:

First, add 8 to both sides: 2x^2 = 8. Then divide by 2: x^2 = 4. Finally, take the square root: x = ±2, so the solutions are x = 2 and x = -2.

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Q5. What is the solution to the inequality 5x + 3 > 2x + 12?

Solution:

Subtract 2x from both sides: 3x + 3 > 12. Then subtract 3: 3x > 9. Finally, divide by 3: x > 3.

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Q6. What is the solution set for the inequality x + 2 > 3?

Solution:

To solve x + 2 > 3, subtract 2 from both sides: x > 1.

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Q7. Factor the quadratic expression x^2 - 5x + 6.

Solution:

To factor x^2 - 5x + 6, we need two numbers that multiply to 6 and add to -5. The numbers -2 and -3 work. Thus, the factorization is (x - 2)(x - 3).

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Q8. Factor the polynomial 2x^2 - 8.

Solution:

Factor out the common term 2: 2(x^2 - 4) = 2(x - 2)(x + 2).

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Q9. Factor the expression x^2 - 4.

Solution:

The expression x^2 - 4 is a difference of squares. It factors to (x - 2)(x + 2).

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Q10. Solve the inequality 3x - 7 < 2.

Solution:

To solve 3x - 7 < 2, add 7 to both sides: 3x < 9. Then divide by 3: x < 3.