If the polynomial [tex]P(x)=27 x^3+9 x^2-3 x-10[/tex] is divided by [tex]3 x-2[/tex], the remainder will be
A) 3
B) 1
C) 0
D) 2
Answer (2)
Use the Remainder Theorem to find the remainder when P ( x ) is divided by 3 x − 2 .
Set the divisor equal to zero: 3 x − 2 = 0 , and solve for x : x = 3 2 .
Evaluate P ( 3 2 ) = 27 ( 3 2 ) 3 + 9 ( 3 2 ) 2 − 3 ( 3 2 ) − 10 .
Simplify the expression to find the remainder: 0 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = 27 x 3 + 9 x 2 − 3 x − 10 and asked to find the remainder when it is divided by 3 x − 2 . We can use the Remainder Theorem to solve this problem. The Remainder Theorem states that if we divide a polynomial P ( x ) by x − c , the remainder is P ( c ) . In our case, we are dividing by 3 x − 2 , so we need to find the value of x that makes 3 x − 2 = 0 .
Finding the Value of x To find the value of x , we set the divisor equal to zero and solve for x :
3 x − 2 = 0 3 x = 2 x = 3 2
Evaluating the Polynomial Now we need to evaluate the polynomial P ( x ) at x = 3 2 :
P ( 3 2 ) = 27 ( 3 2 ) 3 + 9 ( 3 2 ) 2 − 3 ( 3 2 ) − 10
Simplifying the Expression Let's simplify the expression: P ( 3 2 ) = 27 ( 27 8 ) + 9 ( 9 4 ) − 3 ( 3 2 ) − 10 P ( 3 2 ) = 8 + 4 − 2 − 10 P ( 3 2 ) = 12 − 12 P ( 3 2 ) = 0
Conclusion The remainder when P ( x ) is divided by 3 x − 2 is 0.
Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and other structures. The Remainder Theorem can help determine if a certain value is a root of the polynomial, which can be useful in finding the points where a curve intersects the x-axis. In real life, this could represent finding the equilibrium points in a system or the optimal values in a design.
The remainder when the polynomial P ( x ) = 27 x 3 + 9 x 2 − 3 x − 10 is divided by 3 x − 2 is 0 . This is determined using the Remainder Theorem by evaluating the polynomial at x = 3 2 . The answer choice is C.
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