Quadratic Polynomials

Roots

The solutions of the equation ax² + bx+ c = 0 are called the roots (or zeros) of the polynomial ax² + bx+ c.

We have seen in the section on x-intercepts that if c or b is zero then the roots can be found quite easily.

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.

Example 8.

Write x² + 10x+ 21 = (x+h)(x+k)

h×k=21> 0 and h + k =10 > 0
so h > 0 and k > 0.

Possible answers:

h	1	3
k	21	7
h+k	22	10

Hence h =3 and k =7.

So now we can write x² + 10x+ 21 = (x+3)(x+7)

More Exercises (this will open in another window)

 

Exercise 8.

Now try a few:

Write x2 + x + = (x + h)(x + k), where h, k are integers.

h × k = h +k = h and k have / signs. Hence, (x + )(x + )

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.

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