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Question 23

Let $$a,b,c$$ be real numbers such that the polynomial $$f(x)=x^3+ax^2+x+10$$ has three distinct roots and each root of $$f(x)$$ is also a root of the polynomial $$h(x)=x^4+x^3+bx^2+13x+c$$. Then $$h(1)$$ is


Correct Answer: 40

Solution

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Since every root of the cubic $$f$$ is a root of the quartic $$h$$, polynomial division gives $$h(x)=(x+10/c)f(x)$$. Comparing coefficients gives $$c=30$$, $$a=-2$$ and $$b=-5$$. Therefore $$h(1)=1+1-5+13+30=40$$.

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