Graph the polynomial \(p(x)=(x-1)\left(x^2+3 x+1\right)\), clearly identifying all axis intercepts. (3 marks)
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\(p(x)=(x-1)\left(x^2+3 x+1\right)\)
\(\text{Zeros:} \ \ x=1, x=\dfrac{-3 \pm \sqrt{9-4 \cdot 1 \cdot 1}}{2}=\dfrac{-3 \pm \sqrt{5}}{2}\ \ (\approx-2.62,-0.38)\)
\(\text{At}\ \ x=1, m(\text{multiplicity})=1 \ \Rightarrow \ \text{curve crosses}\ x\text {-axis}\)
\(\text{At} \ \ x=\dfrac{-3 \pm \sqrt{5}}{2}, m=1 \ \Rightarrow \ \text{curve crosses} \ x\text {-axis}\)
\(p(0)=(-1)(1)=-1\)