2. factor f(x) = x4 + 10x3 + 35x2 + 50x + 24 completely showing all work and steps with synthetic division. then sketch the graph. use synthetic division

We need to use rational root theorem to find out roots here.

The rational root theorem states that if p(x) is a polynomial with integer coefficients and if is a zero of p(x) then p is a factor of constant term and q is a factor of leasing term coefficient.

Here factors of constant term are 1,2,3,4,6,8,12,24,-1,-2,-3,-4,-6,-8,-12, and -24.

And factors of leading coefficient is -1,1.

Hence possible roots may be -1,1,-2,2,-3,3,-4,4,-6,6,-8,8,-12,12,-24 and 24.

Let us plugin these in f(x) to find zeroes.

Hence x=-1 is a zero which means x-(-1)=x+1 is a factor.

Let us use synthetic division to find quotient.

-1 | 1  10  35  50  24

  | 0  -1  -9  -26   -24

    1    9  26  24    0

Hence quotient is

Since all coefficients are positive, root must be negative. Let's plugin all remaining negative numbers in the quotient.

Hence x+2 is another factor.

Let us find quotient again using synthetic division.

-2 | 1   9  26   24

   | 0  -2  -14   -24

     1    7    12     0

Hence quotient is

Again we got quotient with all positive coefficients, let us plugin remaining negative numbers from rational root theorem.

Hence x+3 is also a factor.

Let us find quotient using synthetic division.

-3 | 1  7  12

   | 0 -3  -12

    1    4    0

Hence quotient is x+4.

So,

Please have a look at the graph attached.

can you give me more detail on this! ?

step-by-step explanation: