2. The GCD of two polynomial is (x-7) and their LCM is x^3-10x^2+11x+70.If p(x) = x^2-5x+41 .find q (x)
pls answer this question very important don't give wrong answer
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Answer:
Given : h(a)=a−7 and LCM m(a)=a
3
−10a
2
+11a+70.
p(a)=a
2
−12a+35
=a
2
−7a−5a+35
=a(a−7)−5(a−7)
=(a−7)(a−5)
∴p(a)=(a−7)(a−5) . (i)
Now, m(a)=a
3
−10a
2
+11a+70
=(a−7)(a
2
−3a−10 [Using synthetic division]
=(a−7)(a
2
−5a+2a−10)
=(a−7)(a−5)(a+2)
∴m(a)=(a−7)(a−5)(a+2) (ii)
q(a) will have (a−7) as one of the factor, since h(a)=a−7
From (ii) and (i), q(a) will have (a+2) as one of the factor.
Also, since h(a)⋅m(a)=p(a)⋅q(a)
⟹q(a)=(a−7)(a+2)
Hence, q(a)=a
2
−5a−14
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