What is the polynomial when the quotient of x^4 – 5x^3 + 3x – 15 divided by it results in x^3 + 3?

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To find the polynomial when the quotient of the polynomial x4 – 5x3 + 3x – 15 divided by an unknown polynomial is x3 + 3, we can follow these steps:

First, we can express the division in the form of:

Dividend = Divisor × Quotient + Remainder

In this case:

x4 - 5x3 + 3x - 15 = (Unknown Polynomial) × (x3 + 3) + Remainder

Since we are dividing a fourth-degree polynomial by a polynomial of degree three, we know that the divisor must be a first-degree polynomial (linear), which we’ll denote as ax + b.

Now, we can use polynomial long division:

  1. We divide the leading term of the dividend (x4) by the leading term of the divisor (x3), which gives us x.
  2. Now we multiply the entire divisor (x3 + 3) by x:
    3. x × (x3 + 3) = x4 + 3x
  3. Next, we subtract this result from our original polynomial:
    4. (x4 - 5x3 + 3x - 15) - (x4 + 3x) = -5x3 - 3x - 15
  4. Next, we divide the leading term of the result (-5x3) by the leading term of the divisor again:
    5. -5x3 ÷ x3 = -5
  5. We multiply the divisor by -5:
    6. -5 × (x3 + 3) = -5x3 - 15
  6. Now, we subtract this from the current remainder:
    7. (-5x3 - 3x - 15) - (-5x3 - 15) = -3x

This means that our remainder is -3x.

We can now express our equation as:

x4 - 5x3 + 3x - 15 = (ax + b)(x3 + 3) - 3x

Through polynomial division, we find the polynomial that satisfies our condition:

Thus, the polynomial we are looking for is:

x + 3

In conclusion, the polynomial that results from dividing x4 – 5x3 + 3x – 15 by (x3 + 3) is x + 3. This solution demonstrates the relationship between the dividend, divisor, quotient, and remainder in polynomial division.

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