A polynomial has the equation
\(Q(x)=(x+1)^2(x-2)\left(x^2+2 x-8\right)\)
- Express \(Q(x)\) as a product of linear factors and determine the multiplicity of each of its roots. (2 marks)
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- Hence, without using calculus, draw a sketch of \(y=Q(x)\), showing all \(x\)-intercepts. (2 marks)
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Show Answers Only
a. \(Q(x) = (x+1)^2(x-2)\left(x^2+2 x-8\right) \)
\(\text{Roots:}\)
\(x=-1 \ \ \text{(multiplicity 2)}\)
\(x=2 \ \ \text{(multiplicity 2)}\)
\(x=4 \ \ \text{(multiplicity 1)}\)
b.
Show Worked Solution
| a. | \(Q(x)\) | \(=(x+1)^2(x-2)\left(x^2+2 x-8\right)\) |
| \(=(x+1)^2(x-2)(x-2)(x+4)\) | ||
| \(=(x+1)^2(x-2)^2(x-4)\) |
\(\text{Roots:}\)
\(x=-1 \ \ \text{(multiplicity 2)}\)
\(x=2 \ \ \text{(multiplicity 2)}\)
\(x=4 \ \ \text{(multiplicity 1)}\)
b. \(Q(x) \ \text{degree}=5, \ \text{Leading coefficient}=1\)
\(\text{As} \ \ x \rightarrow-\infty, y \rightarrow-\infty\)
\(\text{As} \ \ x \rightarrow \infty, y \rightarrow \infty\)
\(\text{At}\ \ x=0, \ y=1^2 \times (-2)^2 \times -4 = -16\)