SOLUTION: Let P(x)=x^3-2x^2-13x-10 a.determine whether x-5 is a factor if P(x). b.find another factor of P(x). c.find a complete factorization of P(x). d. Solve the equation P(x)=0.

Question 985473: Let P(x)=x^3-2x^2-13x-10
a.determine whether x-5 is a factor if 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(52787)   (Show Source): You can put this solution on YOUR website!

a. To determine whether x-5 is a factor of P(x), calculate P(5), i.e. simply substitute the value of 5 into the polynomial. You will get

P(5) = = 125 - 50 - 65 - 10 = 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-5 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 = -2 (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 5, -1 and -2.

The solution is completed.

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