How do you divide the expression 4x^3 + 2x^2 + 3x + 4 by x^4?

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To divide the polynomial expression 4x3 + 2x2 + 3x + 4 by x4, we need to perform polynomial long division or simplify the division. Since we are dividing by a monomial, we can simplify the expression by dividing each term of the polynomial by x4.

Here’s how it works step by step:

  1. Write the polynomial: 4x3 + 2x2 + 3x + 4
  2. Divide each term by x4:
    • 4x3 ÷ x4 = 4 ÷ x = 4x-1
    • 2x2 ÷ x4 = 2 ÷ x2 = 2x-2
    • 3x ÷ x4 = 3 ÷ x3 = 3x-3
    • 4 ÷ x4 = 4x-4

Now, we can combine all the results:

4x-1 + 2x-2 + 3x-3 + 4x-4

This is the final result of dividing 4x3 + 2x2 + 3x + 4 by x4. We notice that each term now has a negative exponent, indicating they are fractions with x in the denominator:

4/x + 2/x2 + 3/x3 + 4/x4

This method is straightforward and allows us to simplify the expression efficiently while understanding the impact of dividing by higher powers of x.

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