could someone please help me with this?

Question

Factor the following problem completely. First factor out the greatest common factor, and then factor the remaining trinomial.

 

y⁴+4y³ -21y^2

Answer 1

starting with the equation

 

y^4+4y^3-21y^2  =yyyy+4yyy-21yy

 

What is the first thing you notice? Answer: there is at least one y in each of the terms separated by the + or - signs so this tells us we can factor out at least one y from each of the terms so the equation now looks like

 

y(y^3+4y^2-21y)

 

now what do you see about the equation inside the () ? y^3+4y^2-21y - correct - each of the terms separated by the + or - signs has at least one y multiplier. What did we do last time we saw at least one y multiplier? Correct again, we factored out a y so now the equation becomes

 

y^2(y^2+4y-21)

 

notice the equation inside () is now y^2+4y-21 so now 1 of the 3 terms separated by the + and - signs doesn't have a y multiplier so we are done factoring the common y's out of the equation. Now we just have to factor

 

y^2+4y-21. Hmmm - How are we going to do that?!!!

 

 

Well let's start with the general form of

 

(y+a)(y+b) and multiply these out using the distributive property

 

(y+a)(y+b)=y(y+b)+a(y+b)=y^2+by+ay+ab=y^2+(a+b)y+ab

 

So what does this have to do with  y^2+4y-21?

well what if

 

a+b=4 and ab=-21 then y^2+(a+b)y+ab=y^2+4y-21 - Wow! But there is still more to be done

 

If we are clever, we notice that 21=3*7 and if b=7 then a+7=4 so that a=-3 so we have

y^2+4y-21=(y-3)(y+7) and the total answer is : drum role please ...

 

y^2(y-3)(y+7) - ta da!

 

So in review, the steps to follow for these kinds of problems is

 

1) factor out all common y multipliers until there is one term in equation without a y multiplier

         for this problem the result at this stage was y^2(y^2+4y-21)

2) solve the y^2+(a+b)y+ab by factoring the ab into 2 numbers that satisfy a x b and a+b 

         for this problem the 21=3x7 so a or b must be 3 or 7

3) put the factors together to get the final answers

        for this problem the factors were y^2 , (y-3), & (y+7) so the answer was y^2(y-3)(y+7)

 

 

Answer 2

y⁴ + 4y³ - 21y²

 

Factor the greatest common factor, y², out of each term:

 

y²(y² + 4y - 21)

 

Now factor the quadratic in the parentheses.  You're looking for two numbers that multiply to give -21 and add to +4.  Those two numbers are 7 and -3 (7*(-3) = -21; 7 + (-3) = 4).  Hence the factors are (y+7) and (y-3):

 

y²(y+7)(y-3)

Answer 3

 

y⁴+4y³-21y²

 

1) Factor the common variable (y²)

y² (y²+4y-21)

 

2) Factor the rest

y² (y+7)(y-3)