smc-984-20-Cubics

Functions, 2ADV F1 2023 HSC 4 MC

The graph of a polynomial is shown.

Which row of the table is correct for this polynomial?

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`B`

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`text{By elimination:}`

`text{Double root where graph SP touches}\ xtext{-axis}`

`:.\ C and D\ text{incorrect}`

`(x-c)^2\ text{occurs where}\ x>0`

`:. c>0\ (A\ text{is incorrect})`

`=>B`

The graph of `y = f(x)` is shown.

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

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`C`

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`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`

Factorise fully `3x^2 - 27`. (2 marks)

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`3 (x + 3) (x – 3)`

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`3x^2 – 27`	`= 3 (x^2 – 9)`
	`= 3 (x + 3) (x – 3)`

Which expression is a factorisation of `8x^3 + 27`?

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`D`

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`8x^3 + 27`

COMMENT: Factorising a cubic is only examinable with scaffolding, as provided here by expanding the answer options.

`= (2x)^3 + 3^3`

`= (2x + 3)(4x^2 – 6x + 9)`

`=> D`