To factorize the given polynomials [tex]\( x^2 + x - 6 \)[/tex] and [tex]\( x^2 - 9 \)[/tex] into their respective factors, follow these steps:
### Factorizing [tex]\( x^2 + x - 6 \)[/tex]:
1. Identify the quadratic polynomial: The given polynomial is [tex]\( x^2 + x - 6 \)[/tex].
2. Find two numbers that multiply to the constant term (-6) and add to the coefficient of the linear term (1):
- We need numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\( a \cdot b = -6 \)[/tex] and [tex]\( a + b = 1 \)[/tex].
3. Determine the factors:
- The numbers that satisfy these conditions are 3 and -2, because [tex]\( 3 \cdot (-2) = -6 \)[/tex] and [tex]\( 3 + (-2) = 1 \)[/tex].
4. Express the polynomial as a product of two binomials:
- Using the numbers found, we can write [tex]\( x^2 + x - 6 \)[/tex] as:
[tex]\[
(x + 3)(x - 2)
\][/tex]
Therefore, the factorization of [tex]\( x^2 + x - 6 \)[/tex] is:
[tex]\[
(x + 3)(x - 2)
\][/tex]
### Factorizing [tex]\( x^2 - 9 \)[/tex]:
1. Identify the quadratic polynomial: The given polynomial is [tex]\( x^2 - 9 \)[/tex].
2. Recognize it as a difference of squares:
- [tex]\( x^2 - 9 \)[/tex] can be written as [tex]\( x^2 - 3^2 \)[/tex].
3. Apply the difference of squares formula:
- The difference of squares formula is [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
4. Rewrite the polynomial using the formula:
- In this case, [tex]\( a = x \)[/tex] and [tex]\( b = 3 \)[/tex].
- So, [tex]\( x^2 - 9 \)[/tex] can be expressed as:
[tex]\[
(x - 3)(x + 3)
\][/tex]
Therefore, the factorization of [tex]\( x^2 - 9 \)[/tex] is:
[tex]\[
(x - 3)(x + 3)
\][/tex]
### Conclusion
- The factorization of [tex]\( x^2 + x - 6 \)[/tex] is [tex]\( (x + 3)(x - 2) \)[/tex].
- The factorization of [tex]\( x^2 - 9 \)[/tex] is [tex]\( (x - 3)(x + 3) \)[/tex].
Thus, the factored forms of the given polynomials are:
[tex]\[
(x + 3)(x - 2), \quad (x - 3)(x + 3)
\][/tex]
