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

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Finding the Polynomial

To find the polynomial we need, let’s denote the polynomial as P(x). According to the information provided, we have the expression:

Q(x) = 4x^5 + 3x^3 + 15

Given that the quotient of Q(x) and another polynomial P(x) equals x^3 + 3, we can set up the equation:

Q(x) = P(x) * (x^3 + 3)

To find P(x), we can rewrite the equation as:

P(x) = Q(x) / (x^3 + 3)

Now, we need to perform polynomial long division to divide 4x^5 + 3x^3 + 15 by x^3 + 3.

Step-by-step Division:

  1. Divide the leading term of 4x^5 by the leading term of x^3, which gives us 4x^2.
  2. Multiply 4x^2 by x^3 + 3 to get 4x^5 + 12x^2.
  3. Subtract this from Q(x): 
    4.             (4x^5 + 3x^3 + 15) - (4x^5 + 12x^2) = 3x^3 - 12x^2 + 15
  4. Now, take 3x^3 and divide it by x^33.
  5. Multiply 3 by (x^3 + 3) to get 3x^3 + 9.
  6. Subtract this from the previous result:
    7.             (3x^3 - 12x^2 + 15) - (3x^3 + 9) = -12x^2 + 6
  7. Now, -12x^2 + 6 is the remainder.

Putting It All Together:

The quotient we found is 4x^2 + 3 and the remainder is -12x^2 + 6. Therefore, we can express P(x) as:

P(x) = 4x^2 + 3 + (R(x) / (x^3 + 3)), where R(x) = -12x^2 + 6.

Conclusion:

Thus, the polynomial P(x) that divides Q(x) and gives a quotient of x^3 + 3 as stated is expressed as:

P(x) = 4x^2 + 3

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