- Analyze the polynomial \(P(x) = x^3 - 7x^2 + 13x - 3\) to identify the given factor \((x - 3)\).
- Use synthetic division with the root \(x = 3\) to divide the polynomial \(P(x)\) by \((x - 3)\).
- Calculate the coefficients of the quotient as \(1, -4, 1\), corresponding to the quadratic expression \(x^2 - 4x + 1\).
- Conclude that the other factor is \(\boxed{x^2 - 4x + 1}\).
Explanation
- Analysis and Setup of the Division Problem
To find the other factor of the polynomial \(P(x) = x^3 - 7x^2 + 13x - 3\), given that \((x - 3)\) is a factor, we need to divide \(P(x)\) by \((x - 3)\). We can use synthetic division for this purpose. The root associated with the factor \((x - 3)\) is \(x = 3\). We will use the coefficients of \(P(x)\), which are \(1\), \(-7\), \(13\), and \(-3\).
- Executing Synthetic Division
To perform synthetic division:
- Write down the root \(3\) and the coefficients \(1, -7, 13, -3\).
- Bring down the first coefficient: \(1\).
- Multiply the root (\(3\)) by the carried-down coefficient (\(1\)) to get \(3\). Add this to the next coefficient (\(-7\)): \(3 + (-7) = -4\).
- Multiply the root (\(3\)) by the result (\(-4\)) to get \(-12\). Add this to the next coefficient (\(13\)): \(-12 + 13 = 1\).
- Multiply the root (\(3\)) by the result (\(1\)) to get \(3\). Add this to the final coefficient (\(-3\)): \(3 + (-3) = 0\).
The remainder is \(0\), confirming that \((x - 3)\) is indeed a factor of the polynomial. The coefficients of the quotient are \(1, -4, 1\).
- Identifying the Other Factor
Based on the coefficients calculated in the previous step, the resulting quotient is \(1x^2 - 4x + 1\). Therefore, the polynomial can be factored as:
\[x^3 - 7x^2 + 13x - 3 = (x - 3)(x^2 - 4x + 1)\]
Comparing this to the given options, the correct factor is \(\left(x^2 - 4x + 1\right)\).
Examples
Polynomial division is a fundamental technique in algebra used to simplify expressions and solve equations, much like division of integers. In real-life scenarios, engineers might use polynomial modeling to represent physical systems (such as the trajectory of a projectile or stress on a material), and factoring these polynomials allows them to find key points, such as where an object hits the ground or where the maximum stress occurs, by finding the roots of the expression.