Question

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

Answer 1

Solution: The factor form of the given polynomial is $f(x)=(x+4)(x+3)(x+2)(x+1)$.

Explanation:

To find the factor form of the given polynomial fisrt find the random value of x for which the vlaue of f(x) is 0.

The value of the function f(x) is 0 for $x=-1$, therefore $(x+1)$ is a factor of given function.

Use synthetic method to divide the given polynomial for by $(x+1)$. Write coefficients of polynomial in top line and -1 on left side of the line. Write first element in bottom line then multiply it by -1 and write it in second line below the element second element and the add. The division by synthetic method is given in figure 1.

The bottom line shows the coefficients of the quotient polynomial.

So $f(x)=(x+1)(x^3+9x^2+26x+24)$

Similarly $(x+2)$ is the factor of given polynomial because for x=-2 the value of parenthesis polynomial is 0.

Use synthetic method to divide the parenthesis polynomial for by $(x+2)$. it is shown in figure 2.

So

$f(x)=(x+1)(x+2)(x^2+7x+12)\\f(x)=(x+1)(x+2)(x^2+3x+4x+12)\\f(x)=(x+1)(x+2)(x(x+3)+4(x+3))\\f(x)=(x+1)(x+2)(x+3)(x+4)$

Hence the factor form of the given polynomial is $f(x)=(x+4)(x+3)(x+2)(x+1)$. The graph of the given function is given in figure 3.