P(x)=2 x^4-x^3+2 x^2-6
What is the remainder when $P(x)$ is divided by $(x-2)$ ?
Answer (2)
Apply the Remainder Theorem: the remainder when P ( x ) is divided by ( x − c ) is P ( c ) .
Substitute x = 2 into P ( x ) : P ( 2 ) = 2 ( 2 ) 4 − ( 2 ) 3 + 2 ( 2 ) 2 − 6 .
Simplify the expression: P ( 2 ) = 32 − 8 + 8 − 6 = 26 .
The remainder is 26 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = 2 x 4 − x 3 + 2 x 2 − 6 and we want to find the remainder when P ( x ) is divided by ( x − 2 ) .
Applying the Remainder Theorem According to the Remainder Theorem, if we divide a polynomial P ( x ) by ( x − c ) , the remainder is P ( c ) . In this case, we are dividing by ( x − 2 ) , so c = 2 . Therefore, we need to evaluate P ( 2 ) .
Substituting x = 2 We substitute x = 2 into the polynomial P ( x ) : P ( 2 ) = 2 ( 2 ) 4 − ( 2 ) 3 + 2 ( 2 ) 2 − 6
Simplifying the Expression Now we simplify the expression: P ( 2 ) = 2 ( 16 ) − 8 + 2 ( 4 ) − 6 P ( 2 ) = 32 − 8 + 8 − 6 P ( 2 ) = 24 + 8 − 6 P ( 2 ) = 32 − 6 P ( 2 ) = 26
Final Answer Therefore, the remainder when P ( x ) is divided by ( x − 2 ) is 26.
Examples
Polynomials are used to model curves in engineering and physics. For example, the trajectory of a projectile can be modeled using a quadratic polynomial. The Remainder Theorem can be used to evaluate the polynomial at a specific point, which can be useful in determining the height of the projectile at a certain time. In computer graphics, polynomials are used to create smooth curves and surfaces. The Remainder Theorem can be used to evaluate the polynomial at a specific point, which can be useful in determining the color of a pixel on the screen.
To find the remainder of the polynomial P ( x ) = 2 x 4 − x 3 + 2 x 2 − 6 when divided by ( x − 2 ) , we evaluate P ( 2 ) . The calculation gives us a remainder of 26. Therefore, the answer is 26 .
;
