Note: if you see symbols like ³ or Þ in various places, then the Symbol or WP MathA font is not installed on your computer. The translation is:
¹ = ¹ (not=) | Ö = Ö (check mark) |
- = - (minus) | Þ = Þ (implies) |
[Shorter solution: spot a factoring by grouping of pairs of terms:]
x3 -
2x2 + x - 2
= 0
Þ
x2(x - 2)
+ 1 (x - 2)
= 0
Þ
(x2 + 1) (x - 2)
= 0
and x2 + 1 = 0
Þ
x2 = -1
Þ
x = ± i
Therefore the complete solution set is
{ 2, ± i } |
[Longer, more systematic solution: seek rational roots:]
f (x) =
x3 -
2x2 + x - 2
Þ
any rational roots must be of the form
(one of { ±1, ±2 }) / (one of { ±1 })
= ±1 or ±2 only.
[Note that Descartes rule of signs applied to the pattern of
alternating signs in the coefficients of
Using the remainder theorem to test these possibilities:
f (1) = | 1 - 2 + 1 - 2 ¹ 0 | X |
f (-1) = | -1 - 2 - 1 - 2 ¹ 0 | X |
f (2) = | 8 - 8 + 2 - 2 = 0 | Ö |
2 | 1 | -2 | 1 | -2 |
---|---|---|---|---|
2 | 0 | 2 | ||
1 | 0 | 1 | 0 |
Synthetic division:
1 | 1 | -6 | 8 | 6 | -9 |
---|---|---|---|---|---|
1 | -5 | 3 | 9 | ||
1 | -5 | 3 | 9 | 0 |
OR, using the remainder theorem:
f (1) =
1 - 6 + 8 + 6
- 9 =
15 - 15 = 0
The remainder = 0 .
Therefore x = 1 is a zero of f.
Synthetic division:
-1 | 1 | -6 | 8 | 6 | -9 |
---|---|---|---|---|---|
-1 | 7 | -15 | 9 | ||
1 | -7 | 15 | -9 | 0 |
OR, using the remainder theorem:
f (-1) =
1 + 6 + 8 - 6
- 9 =
15 - 15 = 0
The remainder = 0 .
Therefore x = -1
is a zero of f.
g(x) = x2 - 6x + 9 |
x = 3, 3 |
End behaviour: a4 = 1 > 0
Þ
curve turns up at both ends.
x-intercepts = zeros = { -1, 1,
3, 3 }
The even multiplicity
Þ
the curve just touches the x axis at x = 3
y-intercept = a0 =
-9
This is sufficient for the following sketch.
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