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# synthetic division

Use synthetic division to decide whether k=2, k=-1, or k=0 are zeros of
f(x)=x^4-6x^3+x^2+24x-20. If not, give the value of f(k)

### 2 Answers by Expert Tutors

Arthur D. | Effective Mathematics TutorEffective Mathematics Tutor
5.0 5.0 (11 lesson ratings) (11)
1
x^4-6x^3+x^2+24x-20

let's try 2

2           1      -6      1      24      -20
2      -8    -14       20

1      -4     -7      10        0
now divide by (x-2)

x^3-4x^2-7x+10
__________________
x-2  / x^4-6x^3+x^2+24x-20
/  x^4-2x^3
________
-4x^3+x^2
-4x^3+8x^2
___________
-7x^2+24x
-7x^2+14x
__________
10x-20
10x-20
_______
0

using x^3-4x^2-7x+10 try 1 using synthetic division

1                  1      -4      -7      10
1       -3     -10

1     -3      -10      0

divide x^3-4x^2-7x+10 by x-1

x^2-3x-10
__________________
x-1 / x^3-4x^2-7x+10
/   x^3-x^2
_________
-3x^2-7x
-3x^2+3x
________
-10x+10
-10x+10
_______
0
factor x^2-3x-10 into (x-5)(x+2)

Why both synthetic division and ordinary division?  They are, for these divisions, equivalent.
You are correct, Michael. By actually dividing you see that the coefficients of the quotient are equal to the results of the synthetic division directly above. The actual division shows that no mistakes were made in the synthetic division process and the number used is really a zero. The student sees the relationship between synthetic division and actual division; that they are, as you said, equivalent.
Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
1
Using synthetic division,
2|1 -6 1 24 -20
......2 -8 -14 20---------------------
..1 -4 -7 10|0

In the same way, you can see k = -1 and k = 0 are not zeros.
f(-1) = -36
f(0) = -20