Question

given the LCM and GCD of the two polynomials are a^3-10^2+11a+70 and and the Polynomials p(x) is a^2-12a+35then find q(x)
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Answer 1

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