smc-750-20-Remainder Theorem

Algebra, MET2 2020 VCAA 2 MC

Let `p(x)=x^{3}-2 a x^{2}+x-1`, where `a \in R`. When `p` is divided by `x+2`, the remainder is 5.

The value of `a` is

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`E`

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`=>E`

The polynomial `p(x) = x^3-ax + b` has a remainder of 2 when divided by `(x-1)` and a remainder of 5 when divided by `(x + 2)`.

Find the values of `a` and `b`. (3 marks)

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`a`	`= 4`
`b`	`= 5`

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`text(Substitute)\ \ b = a+1\ \ text(into)\ \ text{(2)}`

The graph of `P(x) = x^2 + ax + b` cuts the `x`-axis when `x=2.` When `P(x)` is divided by `x + 1`, the remainder is 18.

Find the values of `a` and `b`. (3 marks)

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`a = -7\ \ text(and)\ \ b = 10`

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`P(x) = x^2 + ax + b`

`text(S)text(ince the graph cuts the)\ xtext(-axis at)\ \ x = 2,`

`P(−1) = 18,`

`(−1)^2 − a + b`	`= 18`
`−a + b`	`= 17`	`…\ (2)`

`text(Subtract)\ \ (1) − (2),`

`3a`	`= −21`
`a`	`= −7`

`text(Substitute)\ \ a = −7\ \ text{into (1),}`

`2(−7) + b`	`= −4`
`b`	`= 10`

`:.a = −7\ \ text(and)\ \ b = 10`

For the polynomial `P(x) = x^3 − ax + 4,\ \ P( – 3) = – 5.`

The value of `a` is

A. `− 12`

B. `− 5`

C. `– 3`

D. `3`

E. `6`

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`E`

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`⇒ E`