Question 392539: Factor the following and list all the zeros (roots):
x^3+4x^2-4x-16
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! x^(3)+4x^(2)-4x-16
If a polynomial function has integer coefficients, then every rational zero will have the form (p)/(q) where p is a factor of the constant and q is a factor of the leading coefficient.
p=16_q=1
Find every combination of \(p)/(q). These are the possible roots of the polynomial function.
+-1,+-2,+-4,+-8,+-16
Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is 0, which means it is a root.
(2)^(3)+4(2)^(2)-4(2)-16
Simplify the expression. In this case, the expression is equal to 0 so x=2 is a root of the polynomial.
0
Since 2 is a known root, divide the polynomial by (x-2) to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
(x^(3)+4x^(2)-4x-16)/(x-2)
Complete the synthetic division of ((x^(3)+4x^(2)-4x-16))/((x-2)).
2,1,4,-4,-16:1,1,2,12,16:1,1,6,8,0
Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by 1.
x^(2)+6x+8
Solve the equation to find any remaining roots.
x=-4,-2
The polynomial can be written as a set of linear factors.
(x-2)(x+4)(x+2)
These are the roots (i.e. zeros) of the polynomial x^(3)+4x^(2)-4x-16.
x=2,-4,-2
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