How can I use the remainder theorem to find the remainder of the polynomial px = x^4 + 9x^3 + 5x^2 + 3x + 4 when divided by x – 3?

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To find the remainder of the polynomial p(x) = x^4 + 9x^3 + 5x^2 + 3x + 4 when it is divided by (x – 3), we can apply the Remainder Theorem. The Remainder Theorem states that for any polynomial f(x), the remainder of the division of f(x) by (x – c) is equal to f(c).

In our case, we need to evaluate p(3) because we are dividing by (x – 3).

Now, let’s substitute 3 into the polynomial:

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

Calculating each term individually:

  • (3)^4 = 81
  • 9(3)^3 = 9 imes 27 = 243
  • 5(3)^2 = 5 imes 9 = 45
  • 3(3) = 9
  • 4 = 4

Now add these values together:

p(3) = 81 + 243 + 45 + 9 + 4

Calculating the sum:

p(3) = 81 + 243 = 324
324 + 45 = 369
369 + 9 = 378
378 + 4 = 382

Thus, the remainder when p(x) is divided by (x – 3) is 382.

In summary: The remainder when x^4 + 9x^3 + 5x^2 + 3x + 4 is divided by (x – 3) is 382.

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