if alpha and beta zeroes of the following quadratic polynomial x^2-x-6 then find the value of a^4+B^4
Answers
Answer:
Step-by-step explanation:
Now, if you know α and β , it is very easy to determine the value of the expression α4+β4 . So, how do you find α and β ? Well, you could try looking for simple factors of the quadratic, or you could use the formula for calculating the roots of a quadratic equation or you could form the square. In this particular case, all three methods are very simple.
However, there is a method that you can use for your calculation that doesn’t involving finding α and β ! In this case, it is the more complex way; however, sometimes, it can be quicker.
The trick is to make use of the fact that for the quadratic ax2+bx+c=0 ,
α+β=−ba
αβ=ca
[If α and β are the roots of the quadratic, then a(x−α)(x−β)=0 .
Multiplying this out, we have ax2−a(α+β)x+aαβ=0
Equating the coefficients of this quadratic with the initial quadratic, we have:
−a(α+β)=b→ Equation 1.
aαβ=c→ Equation 2.]
In our quadratic, a=1 , b=−1 and c=−6 , so we have:
α+β=−−11=1
αβ=−61=−6
Now, how does this help us?
Well, α4+β4=(α2+β2)2−2α2β2
=((α+β)2−2αβ)2−2α2β2
=((1)2−2×−6)2−2×(−6)2
=(1+12)2−72=132−72=169−72=97
Answer:
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