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Functions, EXT1 F2 EQ-Bank 1

The polynomial  \(P(x)=2 x^3-m x^2+n x+27\)  has a double root and  \(P(-3)=P^{\prime}(-3)=0\). 

Find the values of \(m\) and \(n\) and hence find the other root of \(P(x)\).   (3 marks)

Show Answers Only

\(m=-15, n=36\)

\(\gamma= -\dfrac{3}{2}\)

Show Worked Solution

\(P(x)=2 x^3-m x^2+n x+27\)

\(P^{′}(x)=6x^2-2m x+n\)

\(P(-3)\) \(=2 \times (-3)^{3}-(-3)^2m-3n+27\)  
\(0\) \(=-54-9m-3n+27\)  
\(9m+3n\) \(=-27\ \ …\ (1)\)  

 

\(P^{′}(-3)\) \(=6 \times (-3)^2+6m+n\)  
\(6m+n\) \(=-54\)  
\(18m+3n\) \(=-162\ …\ (2)\)  

 
   \( (2)-(1):\)

\(9m=-135\ \ \Rightarrow  m=-15\)

\(6 \times -15+n=-54\ \ \Rightarrow \ n=36\)

\(\text{Using product of roots:}\)

\(\alpha \beta \gamma\) \(=\dfrac{-d}{a}\)  
\((-3)^{2} \gamma\) \(=\dfrac{-27}{2} \)  
\(\gamma\) \(=\dfrac{-27}{2 \times 9} = -\dfrac{3}{2}\)  

Functions, EXT1 F2 2023 HSC 14b

Consider the hyperbola  \(y=\dfrac{1}{x}\)  and the circle  \((x-c)^2+y^2=c^2\), where \(c\) is a constant.

  1. Show that the \(x\)-coordinates of any points of intersection of the hyperbola and circle are zeros of the polynomial  \(P(x)=x^4-2 c x^3+1\).  (1 mark)

  2. The graphs of  \(y=x^4-2 c x^3+1\)  for  \(c=0.8\)  and  \(c=1\) are shown.
     

  1. By considering the given graphs, or otherwise, find the exact value of  \(c>0\)  such that the hyperbola  \(y=\dfrac{1}{x}\)  and the circle  \((x-c)^2+y^2=c^2\)  intersect at only one point.  (3 marks)

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  1. \(\text{See Worked Solutions}\)
  2. \(\sqrt[4]{\dfrac{16}{27}}\approx 0.877\)

Show Worked Solution

i.     \(y=\dfrac{1}{x}\ …\ (1) \)

\((x-c)^2+y^2=c^2\ …\ (2) \)

\(\text{Substitute (1) into (2):}\)

\((x-c)^2+\Big{(}\dfrac{1}{x}\Big{)}^2 \) \(=c^2\)  
\(x^2-2cx+c^2+\dfrac{1}{x^2}\) \(=c^2\)  
\(x^4-2cx^3+1\) \(=0\)  
Mean mark (i) 53%.

ii.    \(\text{The two graphs show that for some value of}\ \ 0.8 \leq c \leq 1,\)

\(P(x)\ \text{has a minimum that touches the}\ x\text{-axis once.}\)

\(P(x)\) \(=x^4-2cx^3+1\)  
\(P^{′}(x)\) \(=4x^3-6cx^2\)  

 
\(\text{Find}\ x\ \text{when}\ P^{′}(x)=0: \)

\(4x^3-6cx^2\) \(=0\)  
\(2x^2(2x-3c)\) \(=0\)  
\(x\) \(=\dfrac{3c}{2}\ \ (x \neq 0)\)  

 
\(\text{Find}\ c\ \text{when}\ P(\frac{3c}{2})=0: \)

\(\Big{(} \dfrac{3c}{2} \Big{)}^4-2c\Big{(} \dfrac{3c}{2} \Big{)}^3+1 \) \(=0\)  
\(\dfrac{81c^4}{16}-\dfrac{54c^4}{8}+1\) \(=0\)  
\(\dfrac{(108-81)c^4}{16}\) \(=1\)  
\(\dfrac{27c^4}{16}\) \(=1\)  
\(c^4\) \(=\dfrac{16}{27}\)  
\(c\) \(=\sqrt[4]{\dfrac{16}{27}} \)  
  \(\approx 0.877\)  
Mean mark (ii) 19%.

Functions, EXT1 F2 2022 HSC 13d

The monic polynomial, `P`, has degree 3 and roots `alpha, \beta, \gamma`.

It is given that

           `alpha^(2)+beta^(2)+gamma^(2)=85\ \ and`

           `P^(')(alpha)+P^(')(beta)+P^(')(gamma)=87.`

Find  `alpha beta+beta gamma+gamma alpha`.  (3 marks)

Show Answers Only

`-2`

Show Worked Solution

`P(x)=x^3-(alpha+beta+gamma)x^2+(alphabeta+alphagamma+betagamma)x-alphabetagamma`

`P^(′)(x)=3x^2-2(alpha+beta+gamma)x+alphabeta+alphagamma+betagamma`

`P^(′)(alpha)+P^(′)(beta)+P^(′)(gamma)`

`=3alpha^2-2(alpha+beta+gamma)alpha+alphabeta+alphagamma+betagamma`

       `+ 3beta^2-2(alpha+beta+gamma)beta+alphabeta+alphagamma+betagamma`

       `+ 3gamma^2-2(alpha+beta+gamma)gamma+alphabeta+alphagamma+betagamma`

`=3(alpha^(2)+beta^(2)+gamma^(2))-2(alpha^(2)+beta^(2)+gamma^(2))`

           `-2(alphabeta+alphagamma+alphabeta+betagamma+alphagamma+betagamma)+3(alphabeta+alphagamma+betagamma)`

`=(alpha^(2)+beta^(2)+gamma^(2))-(alphabeta+alphagamma+betagamma)`

`text{Substituting in given values:}`

`85-(alphabeta+alphagamma+betagamma)` `=87`  
`:.alphabeta+alphagamma+betagamma` `=-2`  

♦♦ Mean mark 33%.

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)\ \ 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(As)\ \ x -> ∞, \ p(x) -> ∞`

`=>C`

Functions, EXT1′ F2 2019 HSC 4 MC

The polynomial  `2x^3 + bx^2 + cx + d`  has roots 1 and – 3, with one of them being a double root.

What is a possible value of  `b`?

  1. – 10
  2. – 5
  3. 5
  4. 10
Show Answers Only

`D`

Show Worked Solution

`f(x) = 2x^3 + bx^2 + cx + d`

`f prime (x) = 6x^2 + 2bx + c`

`text(Roots at 1 and −3:)`

`f(1) = 0`

`2 + b + c + d` `= 0`
`b + c + d` `= -2\ \ text{… (1)}`

 
`f(-3) = 0`

`-54 + 9b – 3c + d` `= 0`
`9b – 3c + d` `= 54\ \ text{… (2)}`

 
`(2) – (1)`

`8b – 4c` `= 56`
`2b – c` `= 14\ \ text{… (3)}`

 
`text(If double root at 1:)`

`f prime(1) = 0`

`6 + 2b + c` `= 0`
`2b + c` `= -6\ \ text{… (4)}`

 
`(3) + (4)`

`4b` `= 8`
`b` `= 2`

 
`text(If double root at – 3:)`

`f prime(-3) = 0`

`54 – 6b + c` `= 0`
`-6b + c` `= -54\ \ text{… (5)}`

 
`(3) + (5)`

`-4b` `= -40`
`b` `= 10`

 
`=>   D`

Functions, EXT1′ F2 2018 HSC 11b

The polynomial  `p(x) = x^3 + ax^2 + b`  has a zero at `r` and a double zero at 4.

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

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`a =− 6, b = 32, r = – 2`

Show Worked Solution

`p(x) = x^3 + ax^2 + b`

`pprime(x) = 3x^2 + 2ax`
 

`text(Double root at 4:)`

`pprime(4)` `=0`
`3 xx 16 + 8a` `= 0`
`a` `= −6`

 

`p(4)` `=0`
`64 + 16a + b` `= 0`
`64 – 96 + b` `= 0`
`b` `= 32`

 
`text(Roots of)\ \ p(x)\ \ text(are)\ \ 4, 4, r`

` 4 + 4 +r` `= −a/1 = 6`
`:. r` `= – 2`

Functions, EXT1′ F2 2017 HSC 12d

Let `P(x)` be a polynomial.

  1. Given that  `(x - alpha)^2`  is a factor of `P(x)`, show that
     
    `qquad qquad P(alpha) = P prime (alpha) = 0`.  (2 marks)

  2. Given that the polynomial  `P(x) = x^4 - 3x^3 + x^2 + 4`  has a factor  `(x - alpha)^2`, find the value of `alpha`.  (2 marks)

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  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `2`
Show Worked Solution
i.   `P(x)` `= (x – alpha)^2 · Q(x)`
  `P prime (x)` `= 2 (x – alpha) · Q (x) + (x – alpha)^2 *Q prime (x)`
    `= (x – alpha) [2 Q (x) + (x – alpha) *Q prime (x)]`

 

`P (alpha)` `= 0 xx Q(x) = 0`
`P prime (alpha)` `= 0 [2Q (x) + 0 xx Q prime (x)] = 0`

 
`:. P(alpha) = P prime (alpha) = 0\ text(… as required.)`

 

ii.   `P(x)` `= x^4 – 3x^3 + x^2 + 4`
  `P prime(x)` `= 4x^3 – 9x^2 + 2x`
    `= x (4x^2 – 9x + 2)`
    `= x (4x – 1) (x – 2)`

 
`:. P prime(x) = 0\ \ text(when)\ \ x = 0, 1/4 or 2`

`=>\ text(Multiple roots may exist at)\ \ x=0, 1/4 or 2.`

`text(Test each root in)\ \ P(x):`

`P(0)` `= 0 – 0 + 0 + 4 = 4`
`P(1/4)` `= (1/4)^4 – 3(1/4)^3 + (1/4)^2 + 4= 4 5/256`
`P(2)` `= 16 – 3(8) + 4 + 4 = 0`

 
`:. (x – 2)^2\ \ text(is a factor of)\ \ P(x)`

`:. alpha = 2`

Functions, EXT1′ F2 2016 HSC 13d

Suppose  `p(x) = ax^3 + bx^2 + cx + d`  with `a, b, c` and `d` real, `a != 0.`

  1. Deduce that if  `b^2 - 3ac < 0`  then `p(x)`  cuts the `x`-axis only once.  (2 marks)

  2. If  `b^2 - 3ac = 0 and p(-b/(3a)) = 0`, what is the multiplicity of the root  `x = -b/(3a)?`  (2 marks)

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  1. `text(See Worked Solutions)`
  2. `3`
Show Worked Solution

i.     `p(x) = ax^3 + bx^2 + cx + d`

`pprime(x) = 3ax^2 + 2bx + c`

`=> p(x)\ text(will cut the)\ xtext(-axis once)`

`text(only if)\ \ Delta(pprime(x)) < 0`

`(2b)^2 – 4(3a)c` `< 0`
`4b^2 – 12ac` `< 0`
`b^2 – 3ac` `< 0`

 

ii.   `p(−b/(3a)) = 0`

`pprime(−b/(3a))` `=3a(- b/(3a))^2 + 2b (- b/(3a))+c`
  `=- b^2/(3a)+c`
  `=0\ \ \ (text{given}\ \ b^2 – 3ac = 0)`

 
`:.\ text(Multiplicity at least 2.)`

♦♦ Mean mark 32%.

 
`p″(x) = 6ax + 2b`

`p″(−b/(3a))` `= 6a(−b/(3a)) + 2b=0`

`:. text(Multiplicity of)\ \ x=− b/(3a)\ \ text(is 3.)`

Functions, EXT1′ F2 2016 HSC 2 MC

Which polynomial has a multiple root at  `x = 1?`

  1. `x^5 - x^4 - x^2 + 1`
  2. `x^5 - x^4 - x - 1`
  3. `x^5 - x^3 - x^2 + 1`
  4. `x^5 - x^3 - x + 1`
Show Answers Only

`=> C`

Show Worked Solution

`text(Consider)\ C,`

`P(1)` `= 1 – 1 – 1 + 1 = 0`
`Pprime(x)` `= 5x^4 – 3x^2 – 2x`
`Pprime(1)` `= 5 – 3 – 2 = 0`

 

`:.\ text(Multiple root at)\ x = 1`

`=> C`

Functions, EXT1′ F2 2015 HSC 4 MC

The polynomial  `x^3 + x^2 - 5x + 3`  has a double root at  `x = alpha.`

What is the value of  `alpha ?`

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

`C`

Show Worked Solution
`P prime (x)` `= 3x^2 + 2x – 5`
  `= (3x + 5) (x – 1)`

 
`:. text(Only possible double roots occur when)`

`x=1\ \ text(or)\ \ x=-5/3`
 

`P(1)=1+1-5+3=0`

`:. x = 1\ \ text(is double root.)`

`=>  C`

Functions, EXT1′ F2 2009 HSC 3c

Let  `P(x) = x^3 + ax^2 + bx + 5`, where  `a`  and  `b`  are real numbers.

Find the values of  `a`  and  `b`  given that  `(x - 1)^2`  is a factor of  `P(x).`   (3 marks)

Show Answers Only

`a = 3,\ \ \ b = -9`

Show Worked Solution
`P(x)` `= x^3 + ax^2 + bx + 5`
`P(1)` `= 1 + a + b + 5=0`
 `:.b` `= -a -6\ \ \ \ …\ (1)`

 

`P prime (x)` `= 3x^2 + 2ax + b`
`P prime (1)` `= 3 + 2a + b=0`
 `2a+b` `= -3\ \ \ \ …\ (2)`

 
`text(Substitute)\ \ b=-a-6\ \ text{from  (1)  into  (2)}`

`2a+(-a-6)=-3`

`:.a = 3,\ \ \ b = -9`

Functions, EXT1′ F2 2013 HSC 15b

The polynomial  `P(x) = ax^4 + bx^3 + cx^2 + e`  has remainder `-3` when divided by  `x - 1`. The polynomial has a double root at  `x = -1.`

  1. Show that  `4a + 2c = -9/2.`  (2 marks)

  2. Hence, or otherwise, find the slope of the tangent to the graph  `y = P(x)`  when  `x = 1.`  (1 mark)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `-9`
Show Worked Solution
i.   `P(x)` `= ax^4 + bx^3 + cx^2 + e`
  `P prime (x)` `=4ax^3 + 3bx^2 + 2cx`

 

`P(1)=-3`
`a+b+c+e` `=-3\ \ \ \ …\  (1)`
`P(-1)=0`
`a-b+c+e` `=0\ \ \ \ …\ (2)`
`P′(-1)=0`
`-4a+3b-2c` `=0\ \ \ \ …\ (3)`
   
`(1)-(2)`
`2b` `=-3`
`b` `=-3/2`

 
`text(Substitute into)\ \ (3)`

`:.4a+2c` `=3b`
  `=-9/2\ \ \ \ text(… as required)`

 

ii.  `P prime (1)` `= 4a + 3b + 2c`
  `= -9/2 – 9/2`
  `=-9`

Functions, EXT1′ F2 2014 HSC 14a

Let  `P(x) =x^5 - 10x^2 +15x - 6`.

Show that  `x = 1`  is a root of  `P(x)`  of multiplicity three.  (2 marks)

Show Answers Only

`text(See Worked Solutions.)`

Show Worked Solution

`P(x) =x^5 – 10x^2 +15x – 6`

`P(1) = 1 – 10 + 15 – 6 = 0`
 

`P′(x)` `= 5x^4 – 20x + 15`
`P′(1)` `= 5 – 20 + 15 = 0`
`P″(x)` `= 20x^3 – 20`
`P″(1)` `= 20 – 20 = 0`
`P‴(x)` `= 60x^2`
`P‴(1)` `= 60 ≠ 0`

 
`:.x = 1\ text(is a root of)\ P(x),text(of multiplicity 3.)`

Functions, EXT1 F2 2013 HSC 4 MC

Which diagram best represents the graph  `y = x (1- x)^3 (3- x)^2`?
 

2013 4 mc1

2013 4 mc2

2013 4 mc3

2013 4 mc4

Show Answers Only

`D`

Show Worked Solution

`y = x(1 – x)^3 (3 – x)^2`

`text(By elimination)`

`text(Consider when)\ \ x < 0,`

`y = text{(–ve)} xx text{(+ve)} xx text{(+ve)} < 0`

`:.\ text(Cannot be)\ A\ text(or)\ C`

`text(Consider the cubic factor)\ \ (1 – x)^3,`

`text(The graph must have a stationary point at)\ x = 1`

`:.\ text(Cannot be)\ B`

`=>  D`