Use the remainder theorem to find the remainder when p(x) = x⁴ - 9x³ - 5x² - 3x + 4 is divided by x + 3.

Solution:

Given polynomial p(x) = x⁴ - 9x³ - 5x² - 3x + 4

It is divided by x + 3

For the division, we use remainder theorem

When p(x) is divided by (x-a) then p(a) is the remainder.

p(-3) gives the remainder.

(x⁴ - 9x³ - 5x² - 3x + 4) ÷ (x - 3)

p(x)  = x⁴ - 9x³ - 5x² - 3x + 4

p(-3)  = (-3)⁴ - 9(-3)³ - 5(-3)² - 3(-3) + 4

p(-3)  = 81 - 9(-27) - 5(9) - 3(-3) + 4

p(-3)  = 81 - 9(-27) - 5(9) - 3(-3) + 4

= 81 - 243 - 45 + 13

= 292

The remainder is 292.

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Use the remainder theorem to find the remainder when p(x) = x⁴ - 9x³ - 5x² - 3x + 4 is divided by x + 3.

Summary:

Using the remainder theorem, the remainder when p(x) = x⁴ - 9x³ - 5x² - 3x + 4 is divided by x + 3 is 292.