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Algebra, MET2-NHT 2019 VCAA 3 MC

If  `x + a`  is a factor of  `8x^3 - 14x^2 - a^2 x`, where  `a ∈ R text(\{0})`, then the value of  `a`  is

  1.  7
  2.  4
  3.  1
  4. –2
  5. –1
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`D`

Show Worked Solution
`f(–a)` `= 8(–a)^3 – 14(–a)^2 – a^2(–a)`
`0` `= -8a^3 – 14a^2 + a^3`
`0` `= -7a^3 – 14a^2`
`0` `= -7a^2 (a + 2)`
`a` `= -2`

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 7 MC

If  `x-2`  is a factor of  `2x^3 - 10x^2 + 6x + a`  where  `a in R text{\}{0},`  then the value of `a` is

A.   `-68`

B.   `-20`

C.     `-2`

D.        `2`

E.      `12`

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`E`

Show Worked Solution

`text(S)text(ince)\ \ x-2\ \ text(is a factor,)`

`=> P(2)=0`

`P(2)` `= 2 · 2^3 – 10 · 2^2 + 6 · 2 + a`
`0`  `= 16-40+12+a`
`a` `=12`

 

`=>  E`

Algebra, MET2 2011 VCAA 3 MC

If  `x + a`  is a factor of  `4x^3 - 13x^2 - ax`  where  `a ∈ R text(\{0})`, then the value of `a` is

A.   `−4`

B.   `−3`

C.   `−1`

D.      `1`

E.      `2`

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`=> B`

Show Worked Solution

`text(Let)\ \ p(x) = 4x^3 – 13x^2 – ax`

`text(If)\ \ (x + a)\ \ text(is a factor,)`

`p(−a)` `= 0`
`0` `=4(-a)^3-13(-a)^2-a(-a)`
  `=-4a^3-13a^2+a^2`
  `=-4a^2(a+3)`

 

`:. a = −3,\ \ \ (a != 0)`

`=> B`

Algebra, MET2 2013 VCAA 3 MC

If   `x + a`   is a factor of   `7x^3 + 9x^2 - 5ax`, where  `a in R text(\){0}`, then the value of  `a`  is

A.   `-4`

B.   `-2`

C.   `-1`

D.   `1`

E.   `2`

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`E`

Show Worked Solution

`text(S)text(ince)\ \ x+a\ \ text(is a factor),\ \ f(-a)=0`

`7(-a)^3 + 9(-a)^2 – 5a(-a) = 0,`

`-7a^3 + 14a^2` `=0`
`-7a^2 (a – 2)` `=0`
`:. a = 2,` `\ \ \ a!=0`

 
`=>   E`