How do you find the remainder when the polynomial 4x^3 + 20x + 50 is divided by x^3?

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To find the remainder when the polynomial 4x3 + 20x + 50 is divided by x3, we can use the polynomial remainder theorem. According to this theorem, the remainder of a polynomial division can be found by evaluating the polynomial at specific values.

However, since we are dividing by x3, the remainder will have a degree less than that of x3. This means the remainder will be in the form of a polynomial of degree less than 3, which can be expressed as:

  • R(x) = ax2 + bx + c

where a, b, and c are constants.

For the given polynomial, we can proceed as follows:

  1. First, perform polynomial long division of 4x3 + 20x + 50 by x3.
  2. Since the leading term of the divisor x3 matches that of the highest degree term of the dividend 4x3, we can divide:
  3. The first term yields a quotient of 4, which when multiplied by x3 gives us 4x3.

This means we can write:

4x3 + 20x + 50 = (4)(x3) + (0x2 + 20x + 50)

From this step, we can see that when we subtract 4x3, what remains is:

0x2 + 20x + 50

Since we have already exhausted the highest degree term possible (4x3), the remainder is simply the polynomial that remains:

  • R(x) = 20x + 50

Thus, the remainder when 4x3 + 20x + 50 is divided by x3 is:

20x + 50

In summary, you can find the remainder by performing polynomial long division, and in this case, the answer is the polynomial 20x + 50.

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