Factoring Question Math ?

Question

I just want to check if I did these questions correctly.
1. (2x-y+1)^2 - (x-y-2)^2
2.Determine the x-intercept of the curve by factoring:
y=x^3-4x^2-x+4
Also if you can show how you solved these questions step by step it will help me learn. Thanks

Answer 1

1). (2x - y + 1)² - (x - y - 2)²

Mathematical formula: a² - b² = (a + b)(a - b).

Here a = 2x - y + 1 and b = x - y - 2.

[(2x - y + 1) + (x - y - 2)][(2x - y + 1) - (x - y - 2)]

[2x - y + 1 + x - y - 2][2x - y + 1 - x + y + 2]

[3x - 2y - 1][x + 3]

 

2). y = x³ - 4x² - x + 4

Now the using Synthetic division method:

  1|1  - 4   -1    4

    |0   1   - 3   -4

    |_______________________

-1|1    -3    -4   0

    |0   -1     4

    |_____________________

     1  - 4    0

Now written as factorization is (x - 1)(x + 1)(x - 4)

Therefore y = (x - 1)(x + 1)(x - 4)

y intercepts occur when y = 0

So, (x - 1)(x + 1)(x - 4) = 0.

x - 1 = 0 or x + 1 = 0 or x - 4 = 0

Therefore x = 1 or x = -1 or x = 4.