Question

one factor of the polynomial ( x^3 - 7x^2 + 13x - 3 ) is ( (x - 3) ). what is the other factor of the polynomial?

a. ( (x^2 + 4x - 1) )

b. ( (x^2 - 4x + 1) )

c. ( (x - 4) )

Explanation:

Step1: Perform Polynomial Division

We divide the polynomial \(x^{3}-7x^{2}+13x - 3\) by \((x - 3)\) using long division or synthetic division. Let's use polynomial long division.

Divide \(x^{3}-7x^{2}+13x - 3\) by \(x - 3\):

• First term: \(\frac{x^{3}}{x}=x^{2}\). Multiply \((x - 3)\) by \(x^{2}\) to get \(x^{3}-3x^{2}\). Subtract this from the original polynomial: \((x^{3}-7x^{2}+13x - 3)-(x^{3}-3x^{2})=-4x^{2}+13x - 3\).
• Second term: \(\frac{-4x^{2}}{x}=-4x\). Multiply \((x - 3)\) by \(-4x\) to get \(-4x^{2}+12x\). Subtract this from \(-4x^{2}+13x - 3\): \((-4x^{2}+13x - 3)-(-4x^{2}+12x)=x - 3\).
• Third term: \(\frac{x}{x}=1\). Multiply \((x - 3)\) by \(1\) to get \(x - 3\). Subtract this from \(x - 3\): \((x - 3)-(x - 3)=0\).

So the quotient is \(x^{2}-4x + 1\).

Step2: Verify by Multiplication (Optional)

Multiply \((x - 3)(x^{2}-4x + 1)\):

\[

$$\begin{align*} &(x - 3)(x^{2}-4x + 1)\\ =&x(x^{2}-4x + 1)-3(x^{2}-4x + 1)\\ =&x^{3}-4x^{2}+x-3x^{2}+12x - 3\\ =&x^{3}-7x^{2}+13x - 3 \end{align*}$$

\]

Which matches the original polynomial.

Answer:

B. \((x^{2}-4x + 1)\)