(7) Factorise a $$ x^{3}+x^{2}+x+1 $$ b $$ 2-b^{3}-2 b^{2}+b $$ c $$ x^{2}+p x-q x-p q $$

Question

UpStudy AI · Accepted Answer

Let's factor each expression step by step.

### a) Factor $$ x^{3}+x^{2}+x+1 $$

We can group the terms to factor this expression:

1. Group the first two terms and the last two terms:
   $$
   (x^{3} + x^{2}) + (x + 1)
   $$

2. Factor out the common factors:
   $$
   x^{2}(x + 1) + 1(x + 1)
   $$

3. Now, factor out $$ (x + 1) $$:
   $$
   (x + 1)(x^{2} + 1)
   $$

### b) Factor $$ 2 - b^{3} - 2b^{2} + b $$

Rearranging the terms gives us:
$$
-b^{3} - 2b^{2} + b + 2
$$

1. Factor out $$-1$$:
   $$
   - (b^{3} + 2b^{2} - b - 2)
   $$

2. Now, we can group the terms:
   $$
   - \left( (b^{3} + 2b^{2}) + (-b - 2) \right)
   $$

3. Factor out the common factors:
   $$
   - \left( b^{2}(b + 2) - 1(b + 2) \right)
   $$

4. Factor out $$ (b + 2) $$:
   $$
   - (b + 2)(b^{2} - 1)
   $$

5. Recognize that $$ b^{2} - 1 $$ is a difference of squares:
   $$
   - (b + 2)(b - 1)(b + 1)
   $$

### c) Factor $$ x^{2} + px - qx - pq $$

1. Rearranging gives us:
   $$
   x^{2} + (p - q)x - pq
   $$

2. We can factor this quadratic expression:
   $$
   (x + p)(x - q)
   $$

Now, let's summarize the factored forms:

- a) $$ (x + 1)(x^{2} + 1) $$
- b) $$ - (b + 2)(b - 1)(b + 1) $$
- c) $$ (x + p)(x - q) $$
a) $$ (x + 1)(x^{2} + 1) $$ b) $$ - (b + 2)(b - 1)(b + 1) $$ c) $$ (x + p)(x - q) $$

(7) Factorise
a
b
c

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Solución

a) Factor

b) Factor

c) Factor

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(7) Factorise a b c