The solutions of the equation ax2
+ bx+ c = 0 are called the roots
(or zeros) of the polynomial ax2
+ bx+ c.
Moreover, if we can write the polynomial as (ax – b)(cx – d) then the roots are x =b/a and x = d/c. We consider a simpler case, when a = 1.
Note: (x + h)(x + k) = x2 + (h + k)x + hk
Thus we can factorize x2 + bx + c as (x + h)(x + k) if we can find h, k such that c = h × k and b = x + k.
We are interested in h and k that are integers.
Write x2 + 10x+ 21 = (x+h)(x+k)
h×k=21> 0 and h + k =10
Hence h =3 and k =7.
So now we can write x2 + 10x+ 21 = (x+3)(x+7)
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Not all quadratic polynomials can be written as (x+h)(x+k) where h and k are integers. For example, x2 + 4x +1(x+h)(x+k). For the possible values of h and k are h = 1 and k = 1 but they do not add up to 4.