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# how can i find the remainder theorem and factor theorem to f(x)=4x^6-64x^4+x^3-19

How do I find the remainder theorem and factor theorem of : f(x)=4x^6-64x^4+x^3-19??

### 1 Answer by Expert Tutors

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
1
Possible Rational Roots are : ±1 ,  ± 1 , ±19 , ± 19  , ±19
±2     ±1       ±2     ±4

Polynomial has root of X=1 if and only if Polynomial is divisible by  X-1

Do the long division to determine that:

4X 5 +4X4-60X3-59X2 +59X- 59                                -59X2- 19
X - 1  l  4X6- 64X4+X3- 19                                                      -59X2 -59X
4X6 -4X5                                                                             -59X -19
4X5- 64 X4                                                                        - -59x +59
4X 5- 4 X4                                                                                  -78
-60X4+X3                     f(X) = (X-1) (4X5 +4X4 -60X3-59X2 +59X -59
-60X4+60X3                   then f(1) = -78  the remainder.
-59X3- 19
59X3-59X2           f(1) = 4(1)6- 64(1)4+ (1)3 -19 =-78