Using the remainder theorem, find the remainders obtained when x 3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

By remainder theorem, on dividing f(x) by (x - b), remainder = f(b) ∴ On dividing, f(x) = x 3 + (kx + 8)x + k by x + 1 or (x - (-1)) Remainder = r 1 = f(-1) = -1 3 + ((-1)k + 8)(-1) + k = -1 + (8 - k)(-1) + k = -1 - 8 + k + k = 2k - 9 ∴ On dividing, f(x) = x 3 + (kx + 8)x + k by x - 2 Remainder = r 2 = (2) 3 + (k(2) + 8)2 + k = 8 + 4k + 16 + k = 5k + 24 Given, sum of two remainders = 1 ∴ r 1 + r 2 = 1 The first remainder is 2k - 9 and the second remainder is 5k + 24 and the value of k is -2.

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Mathematics

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

Factorisation

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Mathematics

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

Factorisation

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Mathematics

Using the remainder theorem, find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of two remainders is 1.

Factorisation

Related Questions

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The polynomials ax³ + 3x² - 3 and 2x³ - 5x + a when divided by x - 4 leave the remainder r₁ and r₂ respectively. If 2r₁ = r₂, then find the value of a.

By factor theorem, show that (x + 3) and (2x - 1) are the factors of 2x² + 5x - 3.

Without actual division, prove that x⁴ + 2x³ - 2x² + 2x - 3 is exactly divisible by x² + 2x - 3.