What is the standard form of #y= (x +6)(x + 5)(x -1)^2 #?
1 Answer
Explanation:
Standard form is essentially expanding all the brackets. This converts the equation from factored form to standard form.
In order to make this process easy to understand (especially in this equation with multiple variables), I'll colour code each bracket.
#y=(x+6)(x+5)(x-1)^2#
#y=color(red)((x+6))color(blue)((x+5))color(green)((x-1)^2)#
A squared bracket means that the term is multiplied by itself. In a binomial, you can just FOIL it.
#y=color(red)((x+6))color(blue)((x+5))color(green)((x^2-2x+1)#
Now the first two binomials.
#y=color(purple)((x^2+11x+30))color(green)((x^2-2x+1)#
Now the most complicated part - multiplying two trinomials. I'm going to colour code each bracket as a way to showcase me expanding the brackets. Each colour represents the terms in the first bracket.
#y=color(indigo)(x^4-2x^3+x^2)color(orange)(+11x^3-22x^2+11x)color(blue)(+30x^2-60x+30)#
Now we add like terms.
#y=x^4+9x^3+9x^2-49x+30#
To check our work, I'm going to graph them. Another way is to factor the standard equation and see if the original and new equation are the same.
This is the factored form.
graph{y=(x+6)(x+5)(x-1)^2 [-10, 10, -5, 5]}
This is the standard form.
graph{y=x^4+9x^3+9x^2-49x+30 [-10, 10, -5, 5]}
As you can see, they are the same.
Hope this helps :)
