To factorize the polynomial [tex]\( x^2 - 3x - 40 \)[/tex], let's go through the steps in detail.
Given:
[tex]\[
x^2 - 3x - 40
\][/tex]
1. Identify the coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 1 \)[/tex] (which is [tex]\( a \)[/tex]).
- The coefficient of [tex]\( x \)[/tex] is [tex]\( -3 \)[/tex] (which is [tex]\( b \)[/tex]).
- The constant term is [tex]\( -40 \)[/tex] (which is [tex]\( c \)[/tex]).
2. Set up the middle term factorization:
- We need to find two numbers that multiply to [tex]\( a \cdot c \)[/tex] and add up to [tex]\( b \)[/tex].
- Here, [tex]\( a \cdot c = 1 \cdot (-40) = -40 \)[/tex].
- And [tex]\( b = -3 \)[/tex].
3. Find the pair of factors:
- Consider pairs of factors that multiply to -40:
[tex]\[
(1, -40), (-1, 40), (2, -20), (-2, 20), (4, -10), (-4, 10), (5, -8), (-5, 8)
\][/tex]
- Out of these pairs, the pair that adds up to [tex]\( -3 \)[/tex] is [tex]\( 5 \)[/tex] and [tex]\( -8 \)[/tex].
4. Write the polynomial using these factors:
- Split the middle term using [tex]\( 5 \)[/tex] and [tex]\( -8 \)[/tex]:
[tex]\[
x^2 - 3x - 40 = x^2 + 5x - 8x - 40
\][/tex]
5. Factor by grouping:
- Group the terms:
[tex]\[
(x^2 + 5x) + (-8x - 40)
\][/tex]
- Factor out the common term from each group:
[tex]\[
x(x + 5) - 8(x + 5)
\][/tex]
6. Factor out the common binomial factor:
- From the expression [tex]\( x(x + 5) - 8(x + 5) \)[/tex], we can factor out [tex]\( (x + 5) \)[/tex]:
[tex]\[
(x + 5)(x - 8)
\][/tex]
Thus, the factorized form of the polynomial [tex]\( x^2 - 3x - 40 \)[/tex] is:
[tex]\[
(x - 8)(x + 5)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C \; (x - 8)(x + 5)}
\][/tex]
