When f(x) is divided by x-1, the remainder is 12. When f(x) is divided by x-4, the remainder is 3 . Find the remainder when f(x) is divided by x^(2)-5x+4.

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Understand the Problem: Understand the problem. We are given that f(x) leaves a remainder of 12 when divided by x - 1, and a remainder of 3 when divided by x - 4. We need to find the remainder when f(x) is divided by (x - 1)(x - 4) = x^2 - 5x + 4.Use the Remainder Theorem: Use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by x - a, the remainder is f(a). We apply this theorem to the given information. For x - 1: f(1) = 12 For x - 4: f(4) = 3Express f(x): Express f(x) in terms of the divisors and remainders. Since we know the remainders when f(x) is divided by x - 1 and x - 4, we can express f(x) as: f(x) = (x - 1)Q1(x) + 12 = (x - 4)Q2(x) + 3 where Q1(x) and Q2(x) are some quotient polynomials.Find Relationship: Find the relationship between Q1(x) and Q2(x). Since both (x - 1)Q1(x) + 12 and (x - 4)Q2(x) + 3 represent the same polynomial f(x), and the remainders are given for specific values of x, we can equate the two expressions for those values of x. For x = 1: (1 - 1)Q1(1) + 12 = 12 For x = 4: (4 - 4)Q2(4) + 3 = 3 This does not give us information about Q1(x) and Q2(x), but it confirms the remainders are correct.Find Remainder: Find the remainder when f(x) is divided by (x - 1)(x - 4). We need to find a polynomial R(x) of degree less than 2 (since the divisor is a quadratic) such that: f(x) = (x^2 - 5x + 4)Q(x) + R(x) where Q(x) is the quotient polynomial and R(x) is the remainder polynomial we are looking for.Use Given Remainders: Use the given remainders to determine R(x). Since R(x) must give the same remainders as f(x) for x = 1 and x = 4, we can set up a system of equations: R(1) = 12 R(4) = 3 Assuming R(x) is of the form ax + b, we can write: a(1) + b = 12 a(4) + b = 3Solve Equations: Solve the system of equations for a and b. From the first equation, we get: a + b = 12 From the second equation, we get: 4a + b = 3 Subtracting the first equation from the second, we get: 3a = -9 a = -3 Substituting a back into the first equation: -3 + b = 12 b = 15Write Remainder Polynomial: Write down the remainder polynomial R(x). We found that a = -3 and b = 15, so the remainder polynomial R(x) is: R(x) = -3x + 15

When $f(x)$ is divided by $x-1$ , the remainder is $12$ . When $f(x)$ is divided by $x-4$ , the remainder is $3$ . Find the remainder when $f(x)$ is divided by $x^{2}-5 x+4$ .

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When f(x) f(x) f(x) is divided by x−1 x-1 x−1, the remainder is 121212. When f(x) f(x) f(x) is divided by x−4 x-4 x−4, the remainder is 333 . Find the remainder when f(x) f(x) f(x) is divided by x2−5x+4 x^{2}-5 x+4 x2−5x+4.

When $f(x)$ is divided by $x-1$ , the remainder is $12$ . When $f(x)$ is divided by $x-4$ , the remainder is $3$ . Find the remainder when $f(x)$ is divided by $x^{2}-5 x+4$ .