Question 985638: Let P(x)=x^3-6x^2+5x+12
a. Determine whether x-4 is a factor of P(x).
b. Find another factor of P(x).
c. Find a complete factorization of P(x).
d. Solve the equation P(x)=0.
Answer by ikleyn(52884)   (Show Source):
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a. To determine whether x-4 is a factor of P(x), calculate P(4), i.e. simply substitute the value of 4 into the polynomial. You will get
P(4) =  = 64 - 96 + 20 + 12 = 0.
According wi the Remainder Theorem (see, for example, the lesson Divisibility of polynomial f(x) by binomial x-a in this site),
the binom x-4 is a factor of the polynomial P(x).
b. Make the long division of the polynomial P(x) =  by  . You will get
= . .
So, the polynomial  is another factor of the polynomial P(x).
c. The quadratic polynomial  has the roots  = -1 and  = 3 (use the quadratic formula Introduction into Quadratic Equations
or the Vieta's theorem Solving quadratic equations without quadratic formula, lessons in this site)
It implies that =  . .
Therefore,
P(x) = ..
is the complete factorization of the polynomial P(x) = .
d. Since P(x) = = .., the roots of the polynomial are 4, -1 and 3.
The solution is completed.
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