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\(P(x)\) is a polynomial where \(P(\alpha)=0\) and \(P^{\prime}(\alpha)=0\).
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a. \(P(\alpha)=0\ \ \Rightarrow\ \ (x-a)\ \text{is a factor of \(P(x)\)}\)
\(P(x)=(x-\alpha) \cdot Q(x)\)
\(P^{\prime}(x)=Q(x)+(x-a) \cdot Q(x)\)
\(\text{Since}\ \ P^{\prime}(\alpha)=0:\)
\(Q(\alpha)=0 \ \ \Rightarrow\ \ (x-\alpha) \ \text{is a factor of} \ \ Q(\alpha)\)
\(\text{Let} \ \ Q(x)=(x-\alpha) \cdot R(x)\)
\(P(x)=(x-\alpha)^2 \cdot R(x)\)
\(\therefore \ (x-\alpha)^2 \ \text{is a factor of} \ P(x).\)
b. \(b=6, c=9\)
a. \(P(\alpha)=0\ \ \Rightarrow\ \ (x-a)\ \text{is a factor of \(P(x)\)}\)
\(P(x)=(x-\alpha) \cdot Q(x)\)
\(P^{\prime}(x)=Q(x)+(x-a) \cdot Q(x)\)
\(\text{Since}\ \ P^{\prime}(\alpha)=0:\)
\(Q(\alpha)=0 \ \ \Rightarrow\ \ (x-\alpha) \ \text{is a factor of} \ \ Q(\alpha)\)
\(\text{Let} \ \ Q(x)=(x-\alpha) \cdot R(x)\)
\(P(x)=(x-\alpha)^2 \cdot R(x)\)
\(\therefore \ (x-\alpha)^2 \ \text{is a factor of} \ P(x).\)
b. \(\text{Since the curve is tangent at} \ \ x=-1\)
\(x=-1 \ \ \text{is a double root}\)
\(P(x)=x^3+b x^2+c x+4\)
\(P(-1)=-1+b-c+4=0 \ \ \Rightarrow\ \ b-c=-3\ \ldots\ (1)\)
\(P^{\prime}(x)=3 x^2+26 x+c\)
\(P^{\prime}(-1)=3-2 b+c=0 \ \ \Rightarrow\ \ -2 b+c=-3\ \ldots\ (2)\)
\(\text{Add} \ (1)+(2):\)
\(-b=-6 \ \ \Rightarrow\ \ b=6\)
\(\text{Substitute \(\ b=6\ \) into (1):}\)
\(-6-c=-3 \ \ \Rightarrow\ \ c=9\).
\(\therefore b=6, c=9\)