Certainly! Let's factor the trinomial [tex]\( x^2 + 3x - 4 \)[/tex] step by step.
1. Identify the Trinomial Coefficients:
- The given trinomial is [tex]\( x^2 + 3x - 4 \)[/tex].
- Here, the coefficients are:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = 3 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -4 \)[/tex] (constant term)
2. Find Two Numbers that Multiply to [tex]\( a \cdot c \)[/tex] and Add to [tex]\( b \)[/tex]:
- We need to find two numbers that:
- Multiply to [tex]\( a \cdot c = 1 \cdot (-4) = -4 \)[/tex]
- Add to [tex]\( b = 3 \)[/tex]
3. Determine the Pair of Numbers:
- The pair of numbers [tex]\( m \)[/tex] and [tex]\( n \)[/tex] that satisfy these conditions are:
- [tex]\( m = 4 \)[/tex] and [tex]\( n = -1 \)[/tex]
- These numbers satisfy:
- [tex]\( 4 \times (-1) = -4 \)[/tex]
- [tex]\( 4 + (-1) = 3 \)[/tex]
4. Rewrite the Middle Term Using the Pair of Numbers:
- We rewrite [tex]\( 3x \)[/tex] as [tex]\( 4x - x \)[/tex]:
[tex]\[
x^2 + 4x - x - 4
\][/tex]
5. Factor by Grouping:
- Group the terms in pairs and factor out the common factors:
[tex]\[
x(x + 4) - 1(x + 4)
\][/tex]
6. Factor Out the Common Binomial:
- Notice that [tex]\( x + 4 \)[/tex] is a common factor:
[tex]\[
(x + 4)(x - 1)
\][/tex]
Thus, the factored form of the trinomial [tex]\( x^2 + 3x - 4 \)[/tex] is:
[tex]\[
(x - 1)(x + 4)
\][/tex]
