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# help with polynomial

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When the polynomial p(x) is divided by x - 1, the remainder is 3. When the polynomial p(x)\$ is divided by x - 3, the remainder is 8. What is the remainder when the polynomial p(x) is divided by (x - 1)(x - 3)?

Jun 16, 2021

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According to remainder theorem

When a polynomial p(x) is divided by (x−a), the remainder is p(a).

When p(x) divided by x-1, remainder is 3

... p(1) = 3

When p(x) divided by x-3, remainder is 8

... p(3) = 8

When p(x) is divided by (x-1)(x-3) let remainder be r(x) = ax + b

⇒ p(x) = (x-1)(x-3)q(x) + ax + b

p(1) = a + b = 3

p(3) = 3a + b = 8

Subtracting p(1) from p(3)

2a = 5    ⇒ a = 5/2

⇒ b = 1/2

Thus the remainder is   $$r(x) = {5\over 2}x+{1\over 2}$$

Jun 16, 2021