Sure! Let's factor the given quadratic expressions step-by-step using the Factor Theorem.
### Part (a): [tex]\( x^2 - 6x + 9 \)[/tex]
1. Identify the quadratic expression: [tex]\( x^2 - 6x + 9 \)[/tex]
2. Determine the roots of the equation [tex]\( x^2 - 6x + 9 = 0 \)[/tex]:
- Factoring this quadratic, we look for two numbers that multiply to 9 (constant term) and add to -6 (coefficient of x).
- These two numbers are both -3, because [tex]\(-3 \times -3 = 9\)[/tex] and [tex]\(-3 + (-3) = -6\)[/tex].
3. Write the quadratic as a product of linear factors:
- Since both roots are -3, the expression can be written as [tex]\((x - 3)(x - 3)\)[/tex] or [tex]\((x - 3)^2\)[/tex].
Therefore, the factorized form of [tex]\( x^2 - 6x + 9 \)[/tex] is:
[tex]\[ (x - 3)^2 \][/tex]
### Part (b): [tex]\( x^2 - x - 2 \)[/tex]
1. Identify the quadratic expression: [tex]\( x^2 - x - 2 \)[/tex]
2. Determine the roots of the equation [tex]\( x^2 - x - 2 = 0 \)[/tex]:
- Factoring this quadratic, we look for two numbers that multiply to -2 (constant term) and add to -1 (coefficient of x).
- These two numbers are -2 and +1, because [tex]\(-2 \times 1 = -2\)[/tex] and [tex]\(-2 + 1 = -1\)[/tex].
3. Write the quadratic as a product of linear factors:
- The expression can be written as [tex]\((x - 2)(x + 1)\)[/tex].
Therefore, the factorized form of [tex]\( x^2 - x - 2 \)[/tex] is:
[tex]\[ (x - 2)(x + 1) \][/tex]
### Part (e): [tex]\( x^2 - 5x + 6 \)[/tex]
1. Identify the quadratic expression: [tex]\( x^2 - 5x + 6 \)[/tex]
2. Determine the roots of the equation [tex]\( x^2 - 5x + 6 = 0 \)[/tex]:
- Factoring this quadratic, we look for two numbers that multiply to 6 (constant term) and add to -5 (coefficient of x).
- These two numbers are -2 and -3, because [tex]\(-2 \times -3 = 6\)[/tex] and [tex]\(-2 + (-3) = -5\)[/tex].
3. Write the quadratic as a product of linear factors:
- The expression can be written as [tex]\((x - 2)(x - 3)\)[/tex].
Therefore, the factorized form of [tex]\( x^2 - 5x + 6 \)[/tex] is:
[tex]\[ (x - 2)(x - 3) \][/tex]
In summary:
- The factorized form of [tex]\( x^2 - 6x + 9 \)[/tex] is [tex]\((x - 3)^2\)[/tex].
- The factorized form of [tex]\( x^2 - x - 2 \)[/tex] is [tex]\((x - 2)(x + 1)\)[/tex].
- The factorized form of [tex]\( x^2 - 5x + 6 \)[/tex] is [tex]\((x - 2)(x - 3)\)[/tex].
