TMUA Algebraic Techniques & Manipulation 4

Question 1

If x - 1/x = 3, find x² + 1/x².

Accepted Answer

11

Answer

Squaring x - 1/x = 3 gives x² - 2 + 1/x² = 9, so x² + 1/x² = 11.

Question 2

Find the remainder when x³ - 3x² + 5x - 2 is divided by (x - 2).

Accepted Answer

4

Answer

By the remainder theorem, substitute x=2: 8 - 12 + 10 - 2 = 4.

Question 3

Solve for real x: x/(x-2) + 2/(x+1) = 1

Accepted Answer

x = 0 or x = 3

Answer

Multiplying through by (x-2)(x+1) and simplifying yields x²-3x=0, so x=0 or x=3.

Question 4

Simplify: (√5 + √3)² - (√5 - √3)²

Accepted Answer

4√15

Answer

Using a²-b² = (a+b)(a-b): [(√5+√3)+(√5-√3)][(√5+√3)-(√5-√3)] = (2√5)(2√3) = 4√15.

Question 5

For what values of c does x² + 6x + c > 0 for all real x?

Accepted Answer

c > 9

Answer

The quadratic is always positive when discriminant < 0: 36 - 4c < 0, giving c > 9.

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