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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 (axb)(cxd) 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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