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Polynomials, EXT1 EQ-Bank 5 MC

The graph of  `y = f(x)`  is shown.
 

Which of the following could be the equation of this graph?

  1. `y = (1-x)(2 + x)^3`
  2. `y = (x + 1)(x-2)^3`
  3. `y = (x + 1)(2-x)^3`
  4. `y = (x-1)(2 + x)^3`
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`C`

Show Worked Solution

`text(By elimination:)`

`text(A single negative root occurs when)\ \ x =–1`

`->\ text(Eliminate A and D)`

`text(When)\ \ x = 0, \ y > 0`

`->\ text(Eliminate B)`

`=> C`

Polynomials, EXT1 EQ-Bank 4 MC

Which of the following best represents the graph of  \(y=-5 x(x-2)(3-x)\)?
 

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\(C\)

Show Worked Solution

\(\text{By elimination:}\)

\(\text{Degree = 3,  Leading co-efficient}\ = 5\)

\(\text{As}\ \ x \rightarrow \infty,\ \ y \rightarrow \infty\ \text{(eliminate A and B)}\)

\(\text{When}\ x=1:\)

\(y=-5(-1)(2)=10>0\ \ \text{(eliminate D)}\)

\(\Rightarrow C\)

Functions, EXT1 F2 2020 HSC 5 MC

A monic polynomial `p(x)` of degree 4 has one repeated zero of multiplicity 2 and is divisible by  `x^2 + x + 1`.

Which of the following could be the graph of `p(x)`?

A. B.
C. D.
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`C`

Show Worked Solution

`text(S)text(ince)\ p(x)\ text(is monic,)`

`=> p(x) = (x-a)^2(x^2 + x + 1)`
 

`text(Consider the factor)\ \ (x^2 + x + 1):`

`Delta = sqrt(1^2-4 · 1 · 1) = sqrt(-3) < 0 =>\ text(No roots)`

`:. text(Only root is)\ \ x = a\ \ (text(multiplicity 2))`

`=>\ text(Eliminate)\ \ B and D`
 

`\text{Since}\ p(x)\ \text{is monic:}`

`text(As)\ \ x -> ∞, \ p(x) -> ∞`

`=>C`