# Write as a product of linear factors

Let's try 4: 4 1 -8 17 -4 4 4 1 -4 1 0 P 4 is zero!

### Finding linear factors of polynomials

I'm going to use synthetic division. It is quadratic. Find all real or complex roots of the polynomial. They may be rational, irrational, or imaginary. Since it is not zero, 2 is not a root. It is easy to see if 1 is a root in any polynomial because 1 to any power is 1. If the discriminant is zero, the polynomial has one real root of multiplicity 2.

So x-4 is a factor of P x. The usage of complex numbers makes the statements easier and more "beautiful"!

So the remaining two roots are either irrational or complex. But now we have also observed that every quadratic polynomial can be factored into 2 linear factors, if we allow complex numbers. In the course of a larger application we occassionally need to find the zeros or roots of a polynomial function.

If the discriminant is zero, the polynomial has one real root of multiplicity 2. I'm going to use synthetic division.

### Product of linear factors with complex coefficients

If the discriminant is positive, the polynomial has 2 distinct real roots. But now we have also observed that every quadratic polynomial can be factored into 2 linear factors, if we allow complex numbers. If the discriminant is zero, the polynomial has one real root of multiplicity 2. I'm going to use synthetic division. The "1 -4 1" tells us that is the other factor. The polynomial: has no common factor other than 1 or -1 has too many terms for any of the factoring patterns has too many terms for trinomial factoring will not factor with the "factor by grouping" method. The only factoring technique left is factoring by trial and error of the possible rational roots.

This makes the possible rational roots, 1, -1, 2, -2, 4 or So, this zero must have a multplicity of 2 or 4. So now we're down to 1, 2 or 4 as possible roots.

I think you'll find every term works out negative. Since it is not zero, 2 is not a root. Since we believe that this zero must have a multiplicty of 2 or 4, we will use synthetic division to check if -2 is a zero of the quotient.

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