Question

What is the complete factorization of the polynomial below?

x3 + 4x2 + 16x + 64

A. (x - 4)(x + 4i)(x - 4i)
B. (x + 4)(x + 4i)(x + 4i)
C. (x + 4)(x + 4i)(x - 4i)
D. (x - 4)(x + 4i)(x + 4i)

Answer 1

@Gokuporter @Michele_Laino @rollfaceblah

Answer 2

hint:
if we factor out \(x^2\) between the first and second term, and we factor out \(16\) between the third and fourth term, we get:
\(x^2(x+4)+16(x+4)\)

Answer 3

It is compelling to see the increasing powers of 4 all along the coefficients.

\(4^{0} = 1\)
\(4^{1} = 4\)
\(4^{2} = 16\)
\(4^{3} = 64\)

Thus, one might be willing to believe that x = 4 is a root and (x-4) is a factor. However, as there are NO sign changes in the polynomial, we have to look at -4, instead. Thus, and immediately, throw out A and D. Those both say (x-4) and that cannot be correct.

Test x = -4. Use synthetic division. See what you get.

Answer 4

plug -4 into the equation ?

Answer 5

first prime factor 64 +ve and -ve and then put one by one factor atleast u get zero say x=-4 give zero so its a factor of given polynomial (x+4)

Answer 6

now divide polynomial by x+4 or facror it

Answer 7

please try to factor this expression:
\[\Large {x^2}\left( {x + 4} \right) + 16\left( {x + 4} \right) = \left( {x + 4} \right)(...)\]

Answer 8

If you don't see the factorization provided by ML, then Synthetic Division is the way to go. If you do see the ML factorization, which is easy enough to see right above this post, then simply proceed with your best distributive property.

Answer 9

thank you @Michele_Laino @tkhunny

Answer 10

:)