How to use a grap, synthetic division, and factoring to find the roots of x^3 + 29x + 42 = 12x^2?

Question

How to use a grap, synthetic division, and factoring to find the roots of x^3 + 29x + 42 = 12x^2?

Answer 1

x^3 - 12x^2 + 29x + 42 = 0

Synthetic division:

. . +1 -13 +42

+1)+1 -12 +29 +42

. . . -1 - 1

. . . . . -13 + 29

. . . . . +13 +13

. . . . . . . .  +42 +42

. . . . . . . . . -42 -42

Factoring

(x^2 - 13x + 42) = (x - 6)(x - 7)

x = (-1, 6, 7)

To solve graphically, let

y = x^3 - 12x^2 + 29x + 42

It's easier to calculate in this form:

y = 42 + x(29 - x(12 - x))

Setting up your table for graphing,

. x | 12 - x | (-x=) | +29= | *x= | +42= y

. . | . . . . . /. . . . / . . . . . / . . . . .|

. . | . . . . /. . . . / . . . . . /. . . . . ./

. . | . . . /. . . . / . . . . . / . . . . . /

-7 | 19 | 137 | 162 | -1,134 | -1,092

-6 | 18 | 108 | 137 | -822 | -780

-5 | 17 | 85 | 114 | -570 | -528

-4 | 16 | 64 | 93 | -372 | -330

-3 | 15 | 45 | 74 | -222 | -180

-2 | 14 | 28 | 57 | -114 | -72

-1 | 13 | 13 | 42 | -42 | 0

0 | 12 | 0 | 29 | 0 | 42

1 | 11 | 11 | 18 | 18 | 60

2 | 10 | -20 | 9 | 18 | 60

3 | 9 | -27 | 2 | 6 | 48

4 | 8 | -32 | -3 | -12 | 30

5 | 7 | -35 | -6 | -30 | 12

6 | 6 | -36 | -7 | -42 | 0

7 | 5 | -35 | -6 | -42 | 0

Since the roots appear in the table, you don't even have to graph it out.