Question

Please explain step-by-step on how to factor this function:

952 - 744x + 206x^2 - 24x^3 + x^4

Ans: (28 - 12x + x^2) (34 - 12x + x^2)

I don't get how to arrive at the solution. Thanks.

Answer 1

me neither. were there steps or just the answer? the check is easy enough

Answer 2

myininaya? grouping??

Answer 3

ok maybe we can try something...

Answer 4

\[(x-r_1)(x-r_2)(x-r_3)(x-r_4)=(x^2-(r_1+r_2)x+r_1r_2)(x^2-(r_3+r_4)x+r_3r_4)\]
=>
\[r_1+r_2=12; r_1 \cdot r_2=28\]
also
\[r_3+r_4=12 ; r_3r_4=34\]
wait nvm we aren't suppose to know r_1+r_2=12 and r1*r2=28 and so on...
lol sorry

Answer 5

i don't see how to factor it

Answer 6

well, there are no rational roots to the function

Answer 7

right

Answer 8

Well I figured out a way but it's extremely long and tedious so I was wondering one of you might find something simpler. I geuss there is no other method. :(

Answer 9

what method did you use?

Answer 10

Since, like jamesm said, there are no rational zeros, we know we can't factor a binomial (rational zeros test. Therefore we try to factor it in to 2 quadratic expressions.

We know it has to be like this: (x^2 + Ax + B)(x^2+Cx+D).

Now we expand that group the trms by the exponent:

x^4 + (Cx^3 + Ax^3) + (Dx^2 + ACx^2 + Bx^2) + (ADx + BCx) + (BD)

We can create the following 4 equations:
1. A + C = -24
2. D + AC + B = 206
3. AD + BC = -744
4. BD = 952

The rest can be solved through algebra but it gets very complicated towards the end.

Alternatively, at a point I stopped the algebra, wrote down all the possible factor pairs (there are 8 pairs) for 952 and tried plugging in each one of them. All of them except for 34 & 28 return non-integer values for other coefficients therefore are eliminated.

I really hope there is an easier method because I have to type this up for an assignment in a detailed manner :)

Answer 11

actually i don't know why i didn't think of this method
gj

Answer 12

i can't see an easier way

Answer 13

here is my way:

x^4 - 24x^3 + 206x^2 - 744x + 952 =

you can rewrite it like this:

x^4 - 24x^3 + 144x^2 + 62x^2 - 744x + 952

= (x^2 - 12x)^2 + 62(x^2 - 12x) + 952

Let a = x^2 - 12x

= a^2 + 62a + 952

= (a + 28)(a + 34)