Polynomials again!

Algebra Level 3

A degree 4 polynomial P ( x ) P(x) with leading coefficient 1 has roots of 1, 2 and 3, and k k .

Find P ( 0 ) + P ( 4 ) P(0)+P(4) .


The answer is 24.

Sravanth C. by Sravanth C. , 21, India

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1 solution

Rimson Junio
Aug 27, 2015

We know that P ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − δ ) P(x)=(x-1)(x-2)(x-3)(x-\delta) . Substituting, yields: P ( 0 ) = ( − 1 ) ( − 2 ) ( − 3 ) ( − δ ) = 6 δ P(0)=(-1)(-2)(-3)(-\delta)=6\delta and P ( 4 ) = ( 3 ) ( 2 ) ( 1 ) ( 4 − δ ) = 24 − 6 δ P(4)=(3)(2)(1)(4-\delta)=24-6\delta . Thus P ( 0 ) + P ( 4 ) = 24 P(0)+P(4)=24

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Great observation that the answer is independent of the value of δ \delta .

Did we have to use the fact that the polynomial is monic?

Calvin Lin Staff - 5 years, 9 months ago

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Yes the answer will be of the form 24a where a is the leading coefficient.

Kushagra Sahni - 5 years, 3 months ago

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