simplify the function

Question

f(x)= { 5 x^3 +21 x^2+ 14 x+ 40 }/{ x + 4 }?

Answer 1

f(x) =(5x³ + 21x² + 14x + 40) / (x + 4)

Consider: 5x³ + 21x² + 14x + 40.

If p/q is a rational zero, then p is a factor of 40 and q is a factor of 5.

The possible values of p is -4.

The possible values for q is -4.

So, p/q = -4

Make a table for the synthetic division and test possible real zeros.

p/q	5	21	14	40
-4	5	1	10	0

Since f(-4)   =   0,   x  = - 4 is a zero. The depressed polynomial is  5x² + x + 10 = 0.

Therefore f(x) = (x - 4)(5x² + x + 10) / (x + 4) = 5x² + x + 10

f(x) = 5x² + x + 10.

Comments

Since, f(- 4) =   0, x = - 4 is a zero. The depressed polynomial is  5x² + x + 10 = 0.

So, f(x) = [(x + 4)(5x² + x + 10)]/(x + 4) = 5x² + x + 10.

∴ f(x) = 5x² + x + 10.