Find the value of [tex]$k$[/tex] so that the remainder is 3.
[tex]$(x^2+k x-17) \div(x-2)$[/tex]
A) 8
B) 6
C) 3
D) 1
Answer (2)
Apply the Remainder Theorem: P ( 2 ) = 3 .
Substitute x = 2 into the polynomial: P ( 2 ) = 4 + 2 k − 17 .
Simplify the expression: 2 k − 13 = 3 .
Solve for k : k = 8 .
Explanation
Problem Analysis We are given a polynomial P ( x ) = x 2 + k x − 17 which is divided by x − 2 . The remainder of the division is 3, and we need to find the value of k .
Applying the Remainder Theorem According to the Remainder Theorem, if we divide a polynomial P ( x ) by x − a , the remainder is P ( a ) . In this case, a = 2 , so the remainder is P ( 2 ) . We are given that the remainder is 3, so P ( 2 ) = 3 .
Substitution Substitute x = 2 into P ( x ) : P ( 2 ) = ( 2 ) 2 + k ( 2 ) − 17 = 4 + 2 k − 17 = 2 k − 13
Setting up the Equation Since P ( 2 ) = 3 , we set up the equation: 2 k − 13 = 3
Solving for k Now, we solve for k :
Add 13 to both sides: 2 k = 3 + 13 = 16 Divide by 2: k = 2 16 = 8
Conclusion Therefore, the value of k is 8.
Examples
Polynomials are used to model curves and predict future behavior in various fields. For example, engineers use polynomials to design roads and bridges, ensuring smooth transitions and optimal load distribution. Similarly, economists use polynomials to model economic growth and predict future trends. Understanding the Remainder Theorem allows us to efficiently determine the value of a polynomial at a specific point, which is crucial in these applications.
The value of k for which the remainder is 3 when dividing ( x 2 + k x − 17 ) by ( x − 2 ) is 8. This is determined using the Remainder Theorem, where we substitute 2 into the polynomial, set it equal to 3, and solve for k . The correct answer is A) 8.
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