Calculus, EXT1 EQ-Bank 12

Let \(P(x)=x^3+a x^2+b x-4\) where \(a\) and \(b\) are real numbers.

If \(x=2\) is a double root of \(P(x)=0\), find the values of \(a\) and \(b\). (3 marks)

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\(P(x)=x^3+a x^2+b x-4\)

\(P^{\prime}(x)=3 x^2+2 a x+b\)

\(\text{Since} \ \ x=2\ \ \text{is a double root,}\)

\(P(2)=0:\)

\(8+4 a+2 b-4\)	\(=0\)
\(2 a+b\)	\(=-2\ \ldots\ (1)\)

\(P^{\prime}(2)=0:\)

\(12+4 a+b\)	\(=0\)
\(4 a+b\)	\(=-12\ \ldots\ (2)\)

\([(1) \times 2]-(2):\)

\(b=8\)

\(\text{Substitute} \ \ b=8 \ \ \text{into (1):}\)

\(2a+8\)	\(=-2\)
\(a\)	\(=-5\)