actor the polynomial completely. Do NOT put any spaces in your answers.Must show work/steps on your work page𝑥2−4𝑥+3 =

1. Break Down the Problem

The given polynomial is $x^2 - 4x + 3$. We will factor this quadratic expression completely.

2. Relevant Concepts

To factor a quadratic expression of the form $ax^2 + bx + c$, we need to find two numbers that multiply to $ac$ (the coefficient of $x^2$ times the constant term) and add to $b$ (the coefficient of $x$).

3. Analysis and Detail

1. For the polynomial $x^2 - 4x + 3$, identify $a = 1$, $b = -4$, and $c = 3$.
2. Compute $ac = 1 \times 3 = 3$.
3. We need to find two numbers that multiply to $3$ and add to $-4$. The numbers $-1$ and $-3$ satisfy this condition because:
   • $(-1) + (-3) = -4$
   • $(-1) \times (-3) = 3$
4. Rewrite the middle term using these numbers: $x^2 - 1x - 3x + 3$
5. Group the terms: $(x^2 - 1x) + (-3x + 3)$
6. Factor by grouping: $x(x - 1) - 3(x - 1)$
7. Factor out the common factor $(x - 1)$: $(x - 1)(x - 3)$

4. Verify and Summarize

We can verify our factorization by expanding $(x - 1)(x - 3)$: $x^2 - 3x - 1x + 3 = x^2 - 4x + 3$ The original polynomial is restored, confirming our factorization is correct.

Final Answer

The complete factorization of the polynomial is: $(x-1)(x-3)$