Poly-Factors

Algebra Level 3

Let p ( x ) = x 2 + b x + c p(x) = x^2 + bx + c where b b and c c are integers . If p ( x ) p(x) is a factor of both x 4 + 6 x 2 + 25 x^4 + 6x^2 + 25 and 3 x 4 + 4 x 2 + 28 x + 5 3x^4 + 4x^2 +28x + 5 , then find the value of p ( 1 ) p(1) .

Source : JMO sample paper(2015)


The answer is 4.

Rishu Jaar by Rishu Jaar , 21, India

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1 solution

We have p ( x ) ∣ x 4 + 6 x 2 + 25 p(x)\mid x^4+6x^2+25 , so p ( x ) ∣ 3 x 4 + 18 x 2 + 75 p(x)\mid 3x^4+18x^2+75 .

Combining with p ( x ) ∣ 3 x 4 + 4 x 2 + 28 x + 5 p(x)\mid 3x^4 + 4x^2 +28x + 5 , we get p ( x ) ∣ 14 x 2 − 28 x + 70 p(x)\mid 14x^2-28x+70 or p ( x ) ∣ 14 ( x 2 − 2 x + 5 ) p(x)\mid 14(x^2-2x+5) .

This implies that p ( x ) = x 2 − 2 x + 5 p(x)=x^2-2x+5 .

So, p ( 1 ) = 4 \boxed{p(1)=4} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Amazing solution , much simpler than how I solved it.(+1)

Rishu Jaar - 3 years, 7 months ago

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