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Question 24

If the irreducible quadratic factor of $$5x^4 + 9x^3 - 2x^2 - 4x - 8$$ is $$ax^2 + bx + c$$, then the value of $$a^2 + b^2 - c^2$$ is


Correct Answer: 25

Solution

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Through trial and error we can see that $$x = 1$$ and $$x = -2$$ are roots of the polynomial

We can factorise it as 

$$(x - 1)(x + 2)(5x^2 + 4x + 4)$$.

The quadratic $$5x^2 + 4x + 4$$ has discriminant $$16 - 80 < 0$$, so it is the irreducible factor. 

Hence $$a = 5$$, $$b = 4$$, $$c = 4$$ and 

$$a^2 + b^2 - c^2 = 25 + 16 - 16 = 25$$.

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