Answered

The term $3 x^2$ in the quotient is the result of dividing $3 x^4$ by □ $x^2$.



The polynomial $-2 x^3-x^2+13 x$ in the last line is the result of:

  • dividing $x^2+3 x+1$ by $3 x^2$
  • multiplying $3 x^2$ by $x^2+3 x+1$
  • subtracting $3 x^4+9 x^3+3 x^2$ from the dividend and bringing down $13 x$



Complete the division.



The quotient is $3 x^2+$ □ $x+$ □

Verified Answer :

  • Identify the current remainder as \(-2x^3 - x^2 + 13x\).
  • Divide \(-2x^3\) by \(x^2\) to get the next quotient term, \(-2x\).
  • Subtract \(-2x(x^2 + 3x + 1) = -2x^3 - 6x^2 - 2x\) from the remainder to get \(5x^2 + 15x\).
  • Divide \(5x^2\) by \(x^2\) to get the constant term \(5\).
  • The full quotient is \(3x^2 - 2x + 5\), so the answer is \(\boxed{3x^2 - 2x + 5}\).

Explanation

  1. Problem Analysis and Initial Setup

We are performing polynomial long division. We start with a dividend (implied to be \(3x^4 + 7x^3 + 2x^2 + 13x + C\)) and a divisor \(x^2 + 3x + 1\). The first term of the quotient is \(3x^2\), which is obtained by dividing the leading term of the dividend \(3x^4\) by the leading term of the divisor \(x^2\). Multiplying \(3x^2\) by the divisor \(x^2 + 3x + 1\) yields \(3x^4 + 9x^3 + 3x^2\). Subtracting this from the original dividend gives a remainder of \(-2x^3 - x^2 + 13x\). We now continue the long division process using this new polynomial.

  1. Determining the Second Term of the Quotient

The current remainder to be divided is \(-2x^3 - x^2 + 13x\), and the divisor is \(x^2 + 3x + 1\). To find the next term of the quotient, we divide the leading term of the remainder, \(-2x^3\), by the leading term of the divisor, \(x^2\), which gives \(-2x\). We now multiply this quotient term \(-2x\) by the divisor:
\[ -2x \times (x^2 + 3x + 1) = -2x^3 - 6x^2 - 2x. \]
Next, we subtract this result from our current remainder:
\[ (-2x^3 - x^2 + 13x) - (-2x^3 - 6x^2 - 2x) = (-2x^3 + 2x^3) + (-x^2 + 6x^2) + (13x + 2x) = 5x^2 + 15x. \]

  1. Determining the Final Term of the Quotient

We now have a new remainder of \(5x^2 + 15x\). To find the final term of the quotient, we divide the leading term \(5x^2\) by the divisor's leading term \(x^2\), which gives \(5\). Multiplying this term by the divisor gives:
\[ 5 \times (x^2 + 3x + 1) = 5x^2 + 15x + 5. \]
Subtracting this from our current remainder \(5x^2 + 15x\) yields:
\[ (5x^2 + 15x) - (5x^2 + 15x + 5) = -5. \]
The division is now complete. The quotient is \(3x^2 - 2x + 5\) with a remainder of \(-5\).

Examples

In computer science, when processing algorithms that manipulate large algebraic expressions or simplify complex polynomials (such as in symbolic computation systems), polynomial long division is a fundamental tool. This method is used to divide complex systems of equations or to simplify expressions before performing further calculus-based operations. By breaking down the problem into smaller, manageable steps, we can ensure the precision of the output in any engineering or software design task involving signal processing or data modeling.

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