To solve the given problem step by step:

Problem:

You are given that $(m + n)$ and $(m - n)$ are the factors of the polynomial $x^2 - 3x + 2$. You need to find the value of $m$.

Instructions:

1. (i) Form an equation using factor theorem by substituting $(m + n)$ in place of $x$.

   According to the factor theorem, if $(x - a)$ is a factor of a polynomial, then substituting $x = a$ into the polynomial results in 0.

   The given polynomial is: $x^2 - 3x + 2$ Substituting $x = (m + n)$ into the polynomial: $(m + n)^2 - 3(m + n) + 2 = 0$ Expanding this equation: $m^2 + 2mn + n^2 - 3(m + n) + 2 = 0$ Simplifying: $m^2 + 2mn + n^2 - 3m - 3n + 2 = 0 \quad \text{(Equation 1)}$

2. (ii) Form a second equation by substituting $(m - n)$ in place of $x$.

   Similarly, substitute $x = (m - n)$ into the polynomial: $(m - n)^2 - 3(m - n) + 2 = 0$ Expanding this equation: $m^2 - 2mn + n^2 - 3(m - n) + 2 = 0$ Simplifying: $m^2 - 2mn + n^2 - 3m + 3n + 2 = 0 \quad \text{(Equation 2)}$

3. (iii) From the equations formed in (i) and (ii), find the value of $m$ by equating them.

   Now, subtract Equation 2 from Equation 1: $(m^2 + 2mn + n^2 - 3m - 3n + 2) - (m^2 - 2mn + n^2 - 3m + 3n + 2) = 0$ Simplifying: $4mn - 6n = 0$ Factor the equation: $2n(2m - 3) = 0$ This gives two cases:

   • $n = 0$ (which implies the factors $m + n$ and $m - n$ are the same, not useful for distinct factors),
   • $2m - 3 = 0$, which simplifies to: $m = \frac{3}{2}$

Final Answer:

The value of $m$ is $\frac{3}{2}$.

Would you like more details on any of the steps?

Here are some follow-up questions:

1. How is the factor theorem applied to find factors of a polynomial?
2. Can this method be applied to any degree of polynomial?
3. Why is $n = 0$ not a valid solution in this context?
4. How do we ensure that the substitution method gives the correct value of $m$?
5. What other methods could we use to solve for $m$ besides factor theorem?

Tip: When using the factor theorem, always expand and simplify carefully to avoid small errors in the algebra.