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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)
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`-2`
`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` |
Let `P(x)` be a polynomial.
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i. `text(Proof)\ \ text{(See Worked Solutions)}`
ii. `2`
| i. | `P(x)` | `= (x-alpha)^2 · Q(x)` |
| `P^{′}(x)` | `= 2 (x-alpha) · Q (x) + (x-alpha)^2 *Q^{′}(x)` | |
| `= (x-alpha) [2 Q (x) + (x-alpha) *Q^{′}(x)]` |
| `P (alpha)` | `= 0 xx Q(x) = 0` |
| `P^{′}(alpha)` | `= 0 [2Q (x) + 0 xx Q^{′}(x)] = 0` |
`:. P(alpha) = P^{′}(alpha) = 0\ text(… as required.)`
| ii. | `P(x)` | `= x^4-3x^3 + x^2 + 4` |
| `P^{′}(x)` | `= 4x^3-9x^2 + 2x` | |
| `= x (4x^2-9x + 2)` | ||
| `= x (4x-1) (x-2)` |
`:. P^{′}(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 \frac{5}{256}` |
| `P(2)` | `= 16-3(8) + 4 + 4 = 0` |
`(x-2)^2\ \text(is a factor of)\ P(x)`
`:. alpha = 2`
The polynomial `x^3 + x^2-5x + 3` has a double root at `x = alpha.`
What is the value of `alpha ?`
`C`
`P^{′}(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`