Write the polynomial as a product of linear factors, given that x=a is a zero of P(x)

Question

P(x) = x^3 - (a + b + c)x² + (ab + ac + bc)x - abc

Answer 1

There is a hard way to do this problem and an easy way!

 

The hard way is to actually divide by long division p(x)/x-a.

This will give you a quadratic which you should recognize immediately (or at least be able to factor right away).

 

The other way is to recognize that in an nth degree polynomial the coefficient of the of the term in n-1 is the sum of the roots with the sign changed, the coefficient of the n-2 term is the sum of the the products of the roots taken 2 at a time...and the constant term is the product of the roots * (-1)^n.  These rules come out of the study of the relationship between the roots of a polynomial and the coefficients, which was certainly a part of the study of algebra in the past, but maybe not today.

 

In any event the 3 binomials you need are (x-a)(x-b)(x-c) and you should do the multiplication to convince yourself that this is true.

 

Just as additional information: this same kind of study will show you that if the coefficient of the highest power term is 1, then any integer root divides the constant term.  If it isn't in your book and you want to see that proof, send me a message and I will write it out for you.