The polynomials $a x^3+3 x^2-3$ and $2 x^3-5 x+a$ when divided by $(x-4)$ leave remainders $R_1$ & $R_2$ such that $R_1=R_2$. Then the value of $a$ is
Answer (1)
We are given two polynomials and the remainders when they are divided by ( x − 4 ) are equal. We need to find the value of a .
Use the Remainder Theorem to find R 1 = 64 a + 45 and R 2 = 108 + a .
Set R 1 = R 2 , which gives 64 a + 45 = 108 + a .
Solve for a : 63 a = 63 , so a = 1 .
The value of a is 1 .
Explanation
Understanding the Problem We are given two polynomials, P 1 ( x ) = a x 3 + 3 x 2 − 3 and P 2 ( x ) = 2 x 3 − 5 x + a . When P 1 ( x ) is divided by ( x − 4 ) , the remainder is R 1 . When P 2 ( x ) is divided by ( x − 4 ) , the remainder is R 2 . We are given that R 1 = R 2 . Our objective is to find the value of a . We will use the Remainder Theorem to find the remainders and then equate them to solve for a .
Applying the Remainder Theorem to P1(x) The Remainder Theorem states that if we divide a polynomial P ( x ) by ( x − c ) , the remainder is P ( c ) . Therefore, to find R 1 , we need to evaluate P 1 ( 4 ) .
Calculating R1 We have P 1 ( 4 ) = a ( 4 ) 3 + 3 ( 4 ) 2 − 3 = 64 a + 3 ( 16 ) − 3 = 64 a + 48 − 3 = 64 a + 45 . Thus, R 1 = 64 a + 45 .
Applying the Remainder Theorem to P2(x) Similarly, to find R 2 , we need to evaluate P 2 ( 4 ) .
Calculating R2 We have P 2 ( 4 ) = 2 ( 4 ) 3 − 5 ( 4 ) + a = 2 ( 64 ) − 20 + a = 128 − 20 + a = 108 + a . Thus, R 2 = 108 + a .
Equating R1 and R2 Since R 1 = R 2 , we can set up the equation 64 a + 45 = 108 + a .
Solving for a Now, we solve for a . Subtracting a from both sides gives 63 a + 45 = 108 . Subtracting 45 from both sides gives 63 a = 108 − 45 = 63 . Dividing both sides by 63 gives a = 63 63 = 1 .
Final Answer Therefore, the value of a is 1.
Examples
Polynomials and the Remainder Theorem are useful in various fields, such as engineering and computer science. For example, in signal processing, polynomials can represent signals, and the Remainder Theorem can help determine specific signal characteristics. Imagine you are designing a filter for audio signals. By representing the filter's behavior as a polynomial, you can use the Remainder Theorem to quickly evaluate the filter's response at certain frequencies, allowing you to fine-tune the filter for optimal performance. This ensures that the audio signal is processed correctly, removing unwanted noise and enhancing the desired sounds.
