Question

1. Simplify(√ 3 – 1)3
2. Simplify 4(2n+1 )- 2n+2
--------------------
2n+1 - 2n

3. Given that log2 x + log381 = 1
4. Factorize 6x2 - xy - y2 - 2 x+ y
5. Given that P1(x) = 2x2 + 4x2- x+1

Answer 1

Answer on Question #59133 - Math - Algebra

Question

1. Simplify (∀ 3 - 1)3

Solution

$$\frac {\sqrt {3} - 1}{3} = \frac {\sqrt {3}}{3} - \frac {1}{3} = \frac {1}{\sqrt {3}} - \frac {1}{3}.$$

Answer: $\frac{1}{\sqrt{3}} - \frac{1}{3}$ .

Question

2. Simplify

$$4 (2 n + 1) - 2 n + 2$$

2n+1-2n

Solution

$$\frac {4 (2 n + 1) - 2 n + 2}{2 n + 1 - 2 n} = \frac {8 n + 4 - 2 n + 2}{1} = 6 n + 6 = 6 (n + 1).$$

Answer: $6(n + 1)$

Question

3. Given that $\log 2x + \log 381 = 1$

Solution

$$\log_ {2} x + \log_ {3} 8 1 = 1$$ $$\log_ {3} 8 1 = \log_ {3} (3 ^ {4}) = 4$$ $$\log_ {2} x = 1 - 4 = - 3$$ $$x = 2 ^ {- 3} = \frac {1}{8}.$$

Answer: $x = \frac{1}{8}$ .

Question

4. Factorize $6 \times 2 - xy - y2 - 2x + y$

Solution

$$6 x ^ {2} - x y - y ^ {2} - 2 x + y.$$

We have quadratic polynomial on $x$ and $y$ , so

$$6 x ^ {2} - x y - y ^ {2} - 2 x + y = (a x + b y + c) (d x + e y + f)$$ $$ad = 6; be = -1; ae + bd = -1; cf = 0; af + cd = -2; bf + ce = 1.$$

Let $c = 0$.

$$af = -2; bf = 1 \rightarrow \frac{a}{b} = -2 \rightarrow a = -2b$$

Let $b = 1$:

$$a = -2; d = -\frac{6}{2} = -3; e = -1; f = 1.$$

Thus,

$$6x^2 - xy - y^2 - 2x + y = (y - 2x)(1 - 3x - y).$$

Answer: $(y - 2x)(1 - 3x - y)$

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