Math

• Notes
• HW 2.6
• HW 2.7
• HW 3.1
• HW 3.2
• HW 3.3
• HW 3.4
• HW 3.5
• HW 4.1
• HW 4.2
• HW 4.3
• HW 4.4
• HW 4.5
• Exam November 9th
• Exam November 9th Corrections
• Test Review 03-08-2016

• Chem 7.31 - 7.45

HW 3.4 Q6

$x^4 - 2x^3 - 10x^2 +18x+9=0$

Possible roots $= \pm1, \pm9, \pm3$

Synthetic division:

3| 1  -2  -10  18  9
       3    3 -21 -9
 -------------------
   1   1   -7  -3 |0

$(x-3)(x^3 + x^2 - 7x -3)=0$

Synthetic division:

Possible roots: $\pm1, \pm3$

-3| 1  1  -7  -3
      -3   6   3
 -------------------
    1 -2  -1  |0

$(x+3)(x^2 - 2x -1) = 0$

Apply quadratic formula:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$\frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)}$

$\frac{2 \pm \sqrt{4 +4}}{2} = \frac{2 \pm \sqrt{4 +4}}{2} = \frac{2 \pm \sqrt{8}}{2}$

Simplify the radical:

$\sqrt{8} = \sqrt2 \times \sqrt4 = 2 \times \sqrt2 = 2\sqrt2$

$\frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt2}{2}$

Factor out common terms:

$\frac{2 \pm 2\sqrt2}{2} = \frac{2}{2} \pm \frac{2\sqrt2}{2} = 1 \pm \sqrt2$

$x=3,x = -3, x = 1 - \sqrt2, x = 1+\sqrt2$

HW 3.4 Q9

$8x^3 + 34x^2 - 29x + 5 = 0$

Synthetic division:

-5| 8  34  -29  5
      -40   30 -5
 -------------------
    8  -6    1 |0

$(x+5)(8x^2 -6x + 1) = 0$

Apply quadratic formula:

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$\frac{-(-6) \pm \sqrt{(-6)^2 - 4(8)(1)}}{2(8)}$

$= \frac{6 \pm \sqrt{36 - 32}}{16} = \frac{6 \pm \sqrt{4}}{16} = \frac{6 \pm 2}{16}$

$8/16 = 1/2$
$4/16 = 1/4$

$x=\color{blue}{-5},x=\color{blue}{\frac{1}{2}},x=\color{blue}{\frac{1}{4}}$

HW 3.4 Q19

$12x^3-67x^2+36x-5=0$

Synthetic division:

 1/3|  12  -67  36  -5
             4 -21   5
-------------------
   12  -63  15  |0

$12x^2 -63x + 15$

$\frac{63 \pm \sqrt{63^2 - 4(12)(15)}}{24}$

$= \frac{63 \pm 57}{24}$

$\frac{63-57}{24} = 6/24 = 1/4$

$\frac{63+57}{24} = 120/24 = 60/12 = 30/6 = 15/3 = 5$

HW 3.4 Q20

$2x^3 - 17x^2 -11x -1 =0$

Possible roots: $\pm1, \pm\frac{1}{2}$

Synthetic division:

 -1/2|  2  -17  -11  -1
            -1    9   1
  -------------------
    2  -18   -2  |0

$2x^2 - 18x - 2$

$= \frac{18\pm \sqrt{18^2-4(2)(-2)}}{4}$

$= \frac{18\pm \sqrt{324+16}}{4} = \frac{18\pm \sqrt{340}}{4}$

$\sqrt{340} = \sqrt{2 \times 170} = \sqrt{2 \times 2 \times 85} = 2\sqrt{85}$

$x = \frac{18\pm 2\sqrt{85}}{4} = \color{blue}{\frac{9\pm \sqrt{85}}{2}}$

HW 3.4 Q21

$x^4 - 2x^3 - 47x^2 -64x -20 = 0$

Possible roots: $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$

Synthetic division:

 -1|    1   -2  -47 -64  -20
            -1    3  44   20
  -----------------------
    1   -3  -44 -20   |0

$x^3 - 3x^2 - 44x -20$

Possible roots: $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$

Synthetic division:

 -5|    1   -3  -44 -20
            -5   40  20
  ------------------
    1   -8   -4  |0

$x^2 - 8x -4$

$= \frac{8\pm \sqrt{64+16}}{2} = \frac{8\pm \sqrt{80}}{2}$

$\sqrt{80} = \sqrt{2 \times 40} = 2\sqrt{20} = 4\sqrt{5}$

$x = \frac{8\pm 4\sqrt{5}}{2} = \color{blue}{\frac{4\pm 2\sqrt{5}}{2}}$

Roots: $\brace-5,-1, 4\pm 2\sqrt{5}$

HW 3.4 Q22

$f(x) = 3x^4 - 50x^3 + 160x^2 +290x +77 = 0$

Possible roots: $\pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 3 } , ~ \pm \frac{ 7 }{ 1 } , ~ \pm \frac{ 7 }{ 3 } , ~ \pm \frac{ 11 }{ 1 } , ~ \pm \frac{ 11 }{ 3 } , ~ \pm \frac{ 77 }{ 1 } , ~ \pm \frac{ 77 }{ 3 }$

Synthetic division:

 -1|    3   -50  160  290  77
             -3   53 -213  77
  -----------------------
    3   -53  213   77  |0

$3x^3 - 53x^2 +213x +77$

Synthetic division:

 -1/3|    3   -53  213  77
               -1   18 -77
    ------------------
      3   -54  231  |0

$3x^2 - 54x +231$

$= \frac{54\pm \sqrt{2916-4(3)(231)}}{6} = \frac{54\pm \sqrt{144}}{6}$

$= \frac{54\pm12}{6}$

$x = \frac{66}{6} = 11$

$x = \frac{42}{6} = 7$

Roots: $-1, -\frac{1}{3}, 7,11$

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