The LCM and GCD of a polynomial is (x*4 - y*4) (x*4 + x*2 y*2 + y*4) and x*2 - y*2 . find the q(x) if p(x) = x*4 - y*4
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Asked on January 17, 2020 by
Bappa Síñgh
Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. (x
3
−4x)(5x+1),(5x
2
+x),(5x
3
−9x
2
−2x).
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ANSWER
We know that if p(x) and q(x) are two polynomials, then p(x)×q(x)= {GCD of p(x) and q(x)}× {LCM of p(x) and q(x)}.
Now, it is given that one of the polynomial is p(x)=5x
3
−9x
2
−2x, the LCM is (x
3
−4x)(5x+1) and the GCD is 5x
2
+x, therefore, we have:
(5x
3
−9x
2
−2x)(q(x))=(5x
2
+x)×(x
3
−4x)(5x+1)
⇒[x(5x
2
−9x−2)][q(x)]=[x(5x+1)]×[x(x
2
−4)(5x+1)]
⇒[x(5x
2
−10x+x−2)][q(x)]=x
2
(5x+1)
2
(x
2
−2
2
)
⇒[x(5x(x−2)+1(x−2)][q(x)]=x
2
(5x+1)
2
(x+2)(x−2)(∵(a+b)
2
=a
2
+b
2
+2ab)
⇒[x(5x+1)(x−2)][q(x)]=x
2
(5x+1)
2
(x+2)(x−2)
⇒q(x)=
x(5x+1)(x−2)
x
2
(5x+1)
2
(x+2)(x−2)
⇒q(x)=x(5x+1)(x+2)
Hence, the other polynomial q(x) is x(5x+1)(x+2).