Use the Remainder Theorem to explain whether or not [tex](x-2)[/tex] is a factor of [tex]F(x) = x^4 - 2x^3 + 3x^2 - 10x + 3[/tex].

R ( x ) P ( x ) ​ ∣ Q ( x ) D ( x ) ​
x 4 − 2 x 3 + 3 x 2 − 10 x + 3 ​ ∣ x − 2 ​
− x 4 + 2 x 3 x 4 − 2 x 3 + 3 x 2 − 10 x + 3 ​ ∣ x 3 x − 2 ​
− 3 x 2 + 6 x 3 x 2 − 10 x + 3 ​ ∣ x 3 + 3 x x − 2 ​
4 x − 8 − 4 x + 3 ​ ∣ x 3 + 3 x − 4 x − 2 ​
− 5 ​ ∣ x 3 + 3 x − 4 x − 2 ​
− 5 x 4 − 2 x 3 + 3 x 2 − 10 x + 3 ​ ∣ x 3 + 3 x − 4 x − 2 ​ ​ ​
R ( x ) = − 5

Answered by D3xt3R | 2024-06-10

Using the Remainder Theorem, we find that f ( 2 ) = − 5 , which indicates that ( x − 2 ) is not a factor of F ( x ) . Since the remainder is not zero, we conclude that ( x − 2 ) does not divide F ( x ) evenly. Therefore, ( x − 2 ) is not a factor of the polynomial.
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Answered by D3xt3R | 2024-08-07