How does knowing one linear factor of a polynomial help find the other factors?

How does knowing one linear factor of a polynomial help find the other factors?

Step-by-step explanation:

f(x)=(x−3)(x−1)(x+2)(x+6)

f(x)=(x−2)(x−2)(x+3)(x+5)

f(x)=(x−5)(x−3)(x+2)(x+2)

f(x)=(x−8)(x−1)(x+3)(x+5)

f(x)=(x−2)(x−1)(x+4)(x+4)

Correct

f(x)=(x−2)(x−2)(x+3)(x+5)

Explanation:

We begin by attempting to find any rational roots using the Rational Root Theorem, which states that the possible rational roots are the positive or negative versions of the possible fractional combinations formed by placing a factor of the constant term in the numerator and a factor of the leading coefficient in the denominator.

That was a lot of wordage in one sentence, so let's break that down.  We begin with our polynomial.

f(x)=x4+4x3−13x2−28x+60

The constant term is the term without a variable (just a plain number).  In our case the constant is 60.  What are the possible factors of 60?

1,2,3,4,5,6,10,12,15,20,30,60

The leading coefficient is the number in front of the largest power of the variable.  When the terms are listed in descending order (highest to lowest power), the leading coefficient is always the first number.  In our case the leading coefficient is hard to spot.  Since there is no number in front of x4, the coefficient is 1 by default.

This is nice because the only factor of 1 is well ... 1.

We then create all the possible fractions with a factor of the constant in the numerator and a factor of the leading coefficient in the denominator.  This actually isn't as bad as it could be since our only possible denominator is 1.  Any fraction with a denominator of 1 is just the numerator.  Therfore, our possible "fractions" are simply

1,2,3,4,5,6,10,12,15,20,30,60

However, we must consider the positive or negative versions of these, so our final list of possible rational roots is

±1,±2,±3,±4,±5,±6,±10±,1±2,±15,±20,±30,±60

Unfortunately, this is where the process (at least without the assitance of a graphing calculator) becomes less fun.  Using synthetic division, we must simply try each possible root until we have success. There's really no consistent rule to tell us where to start. Generally starting with the smaller whole numbers is best because the synthetic division is easier.  Therefore, we could begin with 1 then proceed to −1,2,−2, etc.  

For the sake of keeping this explanation as short as possible, I am going to skip straight to 2, where we will first find success.

Therefore, 2 is a root.  However, it is always important to check to see if a root is in fact a double root (it works twice).  Therefore, let's try it one more time.

2 does in fact work twice and is thus a double root.  Since we only have three terms remainng, we can convert from synthetic back to an algebraic expression.

f(x)=x2+8x+15

We can then factor.

f(x)=(x+3)(x+5)

Writing our root of 2 as an algebraic expression gives (x−2).  Since we have double root, we need two of these.  Therfore, our final factored expression is.

f(x)=(x−2)(x−2)(x+3)(x+5)