Answer :
Answer:
Step-by-step explanation:
To determine if ( (x-4) ) is a factor of ( x^4 - 4x^3 - 3x^2 - 2x - 40 ), we can use the Factor Theorem. If ( f(4) = 0 ), then ( (x-4) ) is a factor.
Let’s evaluate ( f(4) ):
( f(4) = 4^4 - 4(4)^3 - 3(4)^2 - 2(4) - 40 ) ( f(4) = 256 - 256 - 48 - 8 - 40 ) ( f(4) = 0 )
Since ( f(4) = 0 ), ( (x-4) ) is indeed a factor of the given polynomial.
To determine whether \( (x - 4) \) is a factor of the polynomial \( x^4 - 4x^3 - 3x^2 - 2x - 40 \), we can use the Factor Theorem. The Factor Theorem states that \( (x - c) \) is a factor of a polynomial \( P(x) \) if and only if \( P(c) = 0 \).
Here, \( c = 4 \). So we need to evaluate the polynomial at \( x = 4 \):
\[
P(x) = x^4 - 4x^3 - 3x^2 - 2x - 40
\]
Let's substitute \( x = 4 \):
\[
P(4) = 4^4 - 4(4^3) - 3(4^2) - 2(4) - 40
\]
Calculate each term:
\[
4^4 = 256
\]
\[
4(4^3) = 4 \cdot 64 = 256
\]
\[
3(4^2) = 3 \cdot 16 = 48
\]
\[
2(4) = 8
\]
Now substitute these values back into the polynomial:
\[
P(4) = 256 - 256 - 48 - 8 - 40
\]
Combine the terms:
\[
P(4) = 256 - 256 - 48 - 8 - 40 = 0 - 48 - 8 - 40 = -96
\]
Since \( P(4) \neq 0 \), we conclude that \( (x - 4) \) is **not** a factor of \( x^4 - 4x^3 - 3x^2 - 2x - 40 \).