Question

Find the product of the following:
(1) (x + 1) (x + 2) (x + 3)
(ii) (x - 2) (x – 3) (x + 4).
​

Answer 1

Answer:

1 (x^2+2x+x+2)(x+3)

(x^2+3x+2)(x+3)

(x^3+3x^2+3x^2+9x+2x+6)

(x^3+6x^2+11x+6)

2 (x^2-2x-3x+6)(x + 4)

(x^2-5x+6)(x + 4)

(x^3+4x^2-5x^2-20x+6x+24)

(x^3-x^2-14x+24)

Answer 2

Answer:

x3 − x2 – 4x + 4 = (x − a)(x − b)(x − c) = 0

Multiplying the brackets together we see that the constant term, 4, must be the number we get when we multiply a, b and c together.

   abc = 4

All the solutions a, b and c must be factors of 4 so there are not many whole numbers that we need to consider.

We have only the following possibilities:

   ±1, ±2 and ±4

We’ll examine each of these numbers to find which ones are solutions of the equation.

   f(1) = 13 − 12 – 4×1 + 4 = 0                        1 is a solution

   f(−1) = (−1)3 − (−1)2 – 4×(−1) + 4 = 6    

   f(2) = 23 − 22 – 4×2 + 4 = 0                         2 is a solution

  f(−2) = (−2)3 − (−2)2 – 4×(−2) + 4 = 0        −2 is a solution

We have now found three solutions so we don’t need to try  4 and −4 as a cubic equation has a maximum of three solutions.

These three numbers give us the values of a, b and c and we can  factorise the equation.

   x3 − x2 – 4x + 4 = (x − 1)(x − 2)(x + 2) = 0