These are smilar, I just don't understand them?

Question

x^4+5x^3-x^2=5x

Answer 1

Your first step is to bring everything over to the left hand side.

x^4 + 5x^3 - x^2 - 5x = 0

Now, factor the biggest term you can out of the 4 terms. In this case, it's x.

x(x^3 + 5x^2 - x - 5) = 0

To factor that cubic, we have to use a method called "grouping". Grouping involves factoring something out of the first two terms, and factoring something out of the last two terms.

x[ x^2 (x + 5) - 1 (x + 5) ] = 0

And now we group them because they both contain an (x+5) term.

x [x^2 - 1] (x + 5) = 0

And we can factor that middle factor as a difference of squares.

x(x-1)(x+1)(x+5) = 0

Now we just equate each factor to 0, giving

x = 0, 1, -1, -5

Answer 2

x^4 + 5x³ - x² = 5x

The secret is to equate to zero and then factorise if possible because if you multiply a set of numbers together and get the answer 0 then one of those numbers HAS TO BE 0

So x^4 + 5x³ - x² - 5x = 0 ← Note that x is a common factor

So x(x³ + 5x² - x - 5) = 0 ← Note that by grouping in pairs (x + 5) becomes a common factor

So x(x²(x + 5) - 1(x + 5)) = 0

So x(x² - 1)(x + 5) = 0 ← Note that x² - 1 is the difference of two squares
ie x² - 1 = x² - 1² and so can be factorised as 'sum by difference'.

So x(x - 1)(x + 1)(x + 5) = 0

ie either x = 0, or x - 1 = 0 or x + 1 = 0 or x + 5 = 0

Whence x = -5, -1, 0 or 1

Answer 3

it's called GROUPING!

x^4 + 5x^3 - x^2 - 5x = 0
x^3(x + 5) - x(x + 5) = 0
(x + 5)(x^3 - x) = 0
x(x+5)(x^2 - 1) = 0
x(x + 5)(x + 1)(x - 1) = 0