Math CentralQuandaries & Queries


Question from David, a student:

Find all the real and imaginary zeros for each polynomial. Factor each polynomial. Leave factors with imaginary zeros in quadratic form.

h(x)= x^5 +2x^4 - 10x^3 -20x^2 +9x + 18

how do i factor each polynomial and what they mean by leaving factors with imaginary zeros in quadratic form can you plz explain? thank you


Factoring this quintic polynomial involves seeing a pattern. If you group the terms

(x5 + 2x4) - (10x3 + 20x2) + (9x + 18)

You can see that each grouping has a factor of x + 2 and extracting this common factor yields

(x5 + 2x4) - (10x3 + 20x2) + (9x + 18) = (x + 2)(ax4 + bx2 + c)

for some numbers a, b and c. This can then be factored again to finally give h(x) as a product of linear factors and hence the zeros are all real.

If the expression had been x3 + 2x2 + 2x + 1 then grouping as (x3 + 1) + (2x2 + 2x) would yield

x3 + 2x2 + 2x + 1 = (x + 1)(x2 + x + 1)

The quadratic x2 + x + 1 has imaginary roots but the instruction "Leave factors with imaginary zeros in quadratic form." means that you should leave the factorization as (x + 1)(x2 + x + 1) and not attempt to factor (x2 + x + 1).



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