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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

  1. `\ \ \ \ 2`
  2. `- 7/4`
  3. `\ \ \ 1/2`
  4. `- 3/2`
  5. `-2`
Show Answers Only

`E`

Show Worked Solution
`P(-2)` `=5`  
`5` `=(-2)^3-2a(-2)^2-2-1`  
`5` `=-8-8a-2-1`  
`8a` `=-16`  
`:.a` `=-2`  

 
`=>E`

Algebra, MET1 SM-Bank 24

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`
Show Worked Solution
`p(x)` `= x^3-ax + b`
`P(1)` `= 2`
`1-a + b` `= 2`
`b` `= a+1\ \ \ …\ text{(1)}`
`P (-2)` `= 5`
`-8 + 2a + b` `= 5`
`2a + b` `= 13\ \ \ …\ text{(2)}`

 

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

`2a + a+1` `= 13`
`3a` `= 12`
`:. a` `= 4`
`:. b` `= 5`

Algebra, MET1 SM-Bank 23

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`

Show Worked Solution

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

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

`P(2)` `=0`  
`2^2 + 2a + b` `= 0`  
`2a + b` `= -4`       `…\ (1)`

 
`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`

Algebra, MET2 SM-Bank 8 MC

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`

Show Worked Solution
`(-3)^3 -a(-3)+4` `= -5`
`-27+3a+4`  `= -5`
`3a`  `=18`
`a`  `= 6`

 
`⇒ E`