To express the product of the binomials [tex]\((3x + 1)(x - 4)\)[/tex] as a trinomial, follow these steps:
1. Identify the terms to be multiplied:
The two binomials are [tex]\((3x + 1)\)[/tex] and [tex]\((x - 4)\)[/tex].
2. Apply the distributive property (also known as the FOIL method for binomials):
Multiply each term in the first binomial by each term in the second binomial.
[tex]\[
(3x + 1)(x - 4) = 3x \cdot x + 3x \cdot (-4) + 1 \cdot x + 1 \cdot (-4)
\][/tex]
3. Perform the individual multiplications:
- [tex]\(3x \cdot x = 3x^2\)[/tex]
- [tex]\(3x \cdot (-4) = -12x\)[/tex]
- [tex]\(1 \cdot x = x\)[/tex]
- [tex]\(1 \cdot (-4) = -4\)[/tex]
4. Combine the like terms:
The expression obtained after multiplication is:
[tex]\[
3x^2 - 12x + x - 4
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-12x + x = -11x
\][/tex]
5. Write the final trinomial:
The trinomial that represents the product of [tex]\((3x + 1)(x - 4)\)[/tex] is:
[tex]\[
3x^2 - 11x - 4
\][/tex]
Therefore, expressing [tex]\((3x + 1)(x - 4)\)[/tex] as a trinomial gives:
[tex]\[
3x^2 - 11x - 4
\][/tex]
