Question:
Let \(a\) be an integer such that all the real roots of the polynomial \(2x^5+5x^4+10x^3+10x^2+10x+10\) lie in the interval \((a,a+1)\). Then, \(|a|\) is equal to ___________.
Let \(a\) be an integer such that all the real roots of the polynomial \(2x^5+5x^4+10x^3+10x^2+10x+10\) lie in the interval \((a,a+1)\). Then, \(|a|\) is equal to ___________.
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\(p'(x)=10(x^2+x+1)^2>0\), so \(p\) is strictly increasing and has exactly one real root; locate it with sign changes.
Updated On: Nov 4, 2026
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Correct Answer: 2
Solution and Explanation
Let \(p(x)=2x^5+5x^4+10x^3+10x^2+10x+10\).
\(p'(x)=10x^4+20x^3+30x^2+20x+10=10(x^4+2x^3+3x^2+2x+1)=10(x^2+x+1)^2>0\) for all real \(x\). So \(p\) is strictly increasing and has exactly one real root.
\(p(-1)=-2+5-10+10-10+10=3>0\) and \(p(-2)=-64+80-80+40-20+10=-34<0\).
The root lies in \((-2,-1)\), so \(a=-2\) and \(|a|=2\).
Answer: 2
\(p'(x)=10x^4+20x^3+30x^2+20x+10=10(x^4+2x^3+3x^2+2x+1)=10(x^2+x+1)^2>0\) for all real \(x\). So \(p\) is strictly increasing and has exactly one real root.
\(p(-1)=-2+5-10+10-10+10=3>0\) and \(p(-2)=-64+80-80+40-20+10=-34<0\).
The root lies in \((-2,-1)\), so \(a=-2\) and \(|a|=2\).
Answer: 2
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