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# Synthetic Division

A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After they explain the concept and process to you, you decide to try and solve some problems.

1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial
a. -3, 2. f(x) = 3x3 + 5x2 - 6x + 18.
a. -4, 2. f(x) = 3x3 + 11x2 - 2x + 8.

2. Given the polynomial f(x) = 2x 3 -5x2- 4x + 3, find all the solutions if the function is completed.
a) f(x) =0
b) f(x+2)=0
d) f(2x) = 0

#### Solution Preview

1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial
a. -3, 2. f(x) = 3x3 + 5x2 - 6x + 18.
a. -4, 2. f(x) = 3x3 + 11x2 - 2x + 8.

1a. -3 3 5 -6 18
&#8595; -9 +12 -18
3 -4 +6 0 = 0 - a root

2 3 5 -6 18
&#8595; +6 +22 +32
3 +11 +16 +50 = +50 - not a root

1b. f(x)=3x^3=11X^2-2x+8

-4 3 11 -2 8
&#8595; -12 4 -8
3 -1 2 0 = 0 - a root

2 3 11 -2 8
&#8595; 6 34 64
3 17 32 72 = 72 - not a root

2. Given the polynomial f(x) = 2x 3 -5x2- 4x + 3, find all the solutions if the function is completed.
a) f(x) =0
b) f(x+2)=0
d) f(2x) = 0

For this problem I have chosen to rational root test

They are the following

a.
f(x) = 0
2x^3-5x^2 - 4x+3=0

Possible solutions +1, + 1/2 , +3, + 3/2

f(1) = 2(1)^3-5(1)-4(1)=3 = 0?
= 2 -5-4+3 = -4 - thus not a solution

f(-1) = 2(-1)^3 -5(-1)^2 -4(01)+3 = 0?
= 2-5+4+3=0
0=0 - thus a solution

f(1/2) = 2(1/2)^3 - 5(1/2)^2 -4(1/2) +3 = 0 ?
= 2/8 - 5/4 - 4/2 = 3 = 0 ?
...

#### Solution Summary

The method of synthetic division is shown and performed on a number of problems. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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