To factor the trinomial [tex]\(7x^2 - 3x - 4\)[/tex], follow these steps:
1. Identify coefficients:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -4\)[/tex]
2. Calculate the product [tex]\(ac\)[/tex]:
- [tex]\( ac = 7 \times (-4) = -28 \)[/tex]
3. Find two numbers that multiply to [tex]\(ac\)[/tex] and add to [tex]\(b\)[/tex]:
- We need two numbers that multiply to [tex]\(-28\)[/tex] and add to [tex]\(-3\)[/tex].
- These numbers are [tex]\(4\)[/tex] (since [tex]\(4 + (-7) = -3\)[/tex] and [tex]\(4 \times (-7) = -28\)[/tex]).
4. Split the middle term using these numbers:
- Rewrite the trinomial: [tex]\(7x^2 - 3x - 4\)[/tex] becomes [tex]\(7x^2 + 4x - 7x - 4\)[/tex].
5. Factor by grouping:
- Group the terms: [tex]\((7x^2 + 4x) + (-7x - 4)\)[/tex].
- Factor out the common factors in each group:
- For the first group: [tex]\(7x(x + \frac{4}{7}) \)[/tex]
- For the second group: [tex]\(-1(7x + 4)\)[/tex].
6. Combine the common factors:
- Notice that both groups have a common factor of [tex]\((x + \frac{4}{7})\)[/tex].
- The factored expression is: [tex]\( (x + \frac{4}{7})(7x - 1) \)[/tex].
However, to see the typical form for integer pairs, it simplifies directly to:
[tex]\((x - 1)(7x + 4)\)[/tex].
So, the factored form of the trinomial [tex]\(7x^2 - 3x - 4\)[/tex] is:
[tex]\[
(x - 1)(7x + 4)
\][/tex]
