Verified Answer :

  • Perform synthetic division on the polynomial \(x^3 - 3x^2 + 5x - 3\) using the root of the divisor \(x-1\), which is \(1\).
  • Calculate the sequence of coefficients for the quotient, which are \(1, -2, 3\).
  • Interpret these coefficients as the quadratic quotient \(x^2 - 2x + 3\).
  • Verify the result by multiplying back, confirming it matches option \(a\), so the quotient is \(\boxed{x^2-2x+3}\).

Explanation

  1. Problem Analysis

We are given the polynomial \(p(x) = x^3 - 3x^2 + 5x - 3\) and we wish to divide it by the linear polynomial \(d(x) = x - 1\). The division of a cubic polynomial by a linear term results in a quadratic quotient \(q(x)\) and a constant remainder \(r\), such that \(p(x) = (x - 1)q(x) + r\). To find the quotient, we can use synthetic division.

  1. Synthetic Division Process

To perform synthetic division by \(x - 1\), we use the root \(x = 1\). We list the coefficients of \(p(x) = x^3 - 3x^2 + 5x - 3\), which are \(1, -3, 5, -3\).

  1. Bring down the leading coefficient \(1\).
  2. Multiply \(1\) by the root \(1\) to get \(1\). Add this to the next coefficient \(-3\): \(-3 + 1 = -2\).
  3. Multiply \(-2\) by the root \(1\) to get \(-2\). Add this to the next coefficient \(5\): \(5 - 2 = 3\).
  4. Multiply \(3\) by the root \(1\) to get \(3\). Add this to the next coefficient \(-3\): \(-3 + 3 = 0\).

The resulting coefficients of the quotient are \(1, -2, 3\), and the remainder is \(0\). Thus, the quotient is \(x^2 - 2x + 3\).

  1. Verification and Conclusion

The quotient obtained through synthetic division is \(q(x) = x^2 - 2x + 3\). We check our result by multiplying the quotient by the divisor and adding the remainder:
\((x - 1)(x^2 - 2x + 3) = x(x^2 - 2x + 3) - 1(x^2 - 2x + 3) = (x^3 - 2x^2 + 3x) - (x^2 - 2x + 3) = x^3 - 3x^2 + 5x - 3\).
This matches our original polynomial perfectly. Comparing our result with the given options, we see that it matches option a.

Examples

Polynomial division is essential in algebra for simplifying expressions, solving equations, and finding roots of polynomials. For example, if you are modeling a physical system where the trajectory of an object follows a cubic equation and you know one point where the object passes through a specific plane (represented by \(x=1\)), you can use this division method to find the simplified quadratic path of the object in the remaining dimensions. Understanding this technique allows for efficient manipulation of complex algebraic models used in engineering and physics.

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