$P(x)=x^4-2 x^3-3 x^2+4$
What is the remainder when $P(x)$ is divided by $(x-3)$?
Answer (2)
Apply the Remainder Theorem: The remainder when P ( x ) is divided by ( x − c ) is P ( c ) .
Evaluate P ( 3 ) to find the remainder: P ( 3 ) = ( 3 ) 4 − 2 ( 3 ) 3 − 3 ( 3 ) 2 + 4 .
Calculate P ( 3 ) = 81 − 54 − 27 + 4 = 4 .
The remainder when P ( x ) is divided by ( x − 3 ) is 4 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = x 4 − 2 x 3 − 3 x 2 + 4 and we want to find the remainder when P ( x ) is divided by ( x − 3 ) .
Applying the Remainder Theorem According to the Remainder Theorem, the remainder when a polynomial P ( x ) is divided by ( x − c ) is P ( c ) . In this case, we want to divide by ( x − 3 ) , so c = 3 . Therefore, we need to evaluate P ( 3 ) .
Substituting x = 3 We substitute x = 3 into the polynomial P ( x ) : P ( 3 ) = ( 3 ) 4 − 2 ( 3 ) 3 − 3 ( 3 ) 2 + 4
Calculating P(3) Now we calculate the value: P ( 3 ) = 81 − 2 ( 27 ) − 3 ( 9 ) + 4 = 81 − 54 − 27 + 4 = 4 So, P ( 3 ) = 4 .
Final Answer Therefore, the remainder when P ( x ) is divided by ( x − 3 ) is 4.
Examples
Polynomials can be used to model various real-world phenomena, such as the trajectory of a projectile or the growth of a population. The Remainder Theorem is useful in determining factors and roots of polynomials, which can help in analyzing these models. For example, if we know the remainder when dividing a polynomial by ( x − a ) is zero, then ( x − a ) is a factor of the polynomial. This can help us find the roots of the polynomial, which represent important values in the model, such as the time when a projectile hits the ground or the population reaches a certain level. Understanding polynomial division and the Remainder Theorem allows us to analyze and make predictions about these real-world situations.
The remainder when the polynomial P ( x ) = x 4 − 2 x 3 − 3 x 2 + 4 is divided by ( x − 3 ) is 4. This is found by evaluating P ( 3 ) using the Remainder Theorem. Thus, the final result is 4 .
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