A polynomial has the equation
\(P(x)=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\).
- By expressing \(P(x)\) as a product of its linear factors, determine the multiplicity of each of the roots of \(P(x)=0\). (2 marks)
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- Hence, without using calculus, draw a sketch of \(y=P(x)\) showing all \(x\)-intercepts. (2 marks)
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Show Answers Only
| a. | \(P(x)\) | \(=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\) |
| \(=(x-1)(x-3)(x+2)^2(x-3)(x+2)\) | ||
| \(=(x-1)(x-3)^2(x+2)^3\) |
\(\text{Roots:}\)
\(x=1\ \text{(multiplicity 1)}\)
\(x=-2\ \text{(multiplicity 3)}\)
\(x=3\ \text{(multiplicity 2)}\)
b.
Show Worked Solution
| a. | \(P(x)\) | \(=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\) |
| \(=(x-1)(x-3)(x+2)^2(x-3)(x+2)\) | ||
| \(=(x-1)(x-3)^2(x+2)^3\) |
\(\text{Roots:}\)
\(x=1\ \text{(multiplicity 1)}\)
\(x=-2\ \text{(multiplicity 3)}\)
\(x=3\ \text{(multiplicity 2)}\)
b.