Of course! Let's take the product of the expressions [tex]\(\left(\frac{2}{3} x + 1\right)\)[/tex] and [tex]\(\left(\frac{1}{5} x - \frac{1}{4}\right)\)[/tex] and express it as a trinomial in its simplest form.
First, we need to use the distributive property (also known as the FOIL method for binomials) to multiply these two expressions:
[tex]\[
\left(\frac{2}{3} x + 1\right) \left(\frac{1}{5} x - \frac{1}{4}\right)
\][/tex]
We distribute each term in the first binomial to each term in the second binomial:
Step 1: Multiply [tex]\(\frac{2}{3} x\)[/tex] by each term in the second binomial:
[tex]\[
\frac{2}{3} x \cdot \frac{1}{5} x = \frac{2}{15} x^2
\][/tex]
[tex]\[
\frac{2}{3} x \cdot \left(-\frac{1}{4}\right) = -\frac{2}{12} x = -\frac{1}{6} x
\][/tex]
Step 2: Multiply [tex]\(1\)[/tex] by each term in the second binomial:
[tex]\[
1 \cdot \frac{1}{5} x = \frac{1}{5} x
\][/tex]
[tex]\[
1 \cdot \left(-\frac{1}{4}\right) = -\frac{1}{4}
\][/tex]
Step 3: Add all these products together:
[tex]\[
\frac{2}{15} x^2 - \frac{1}{6} x + \frac{1}{5} x - \frac{1}{4}
\][/tex]
Step 4: Combine the like terms:
The [tex]\(x\)[/tex] terms are [tex]\(-\frac{1}{6} x\)[/tex] and [tex]\(\frac{1}{5} x\)[/tex]:
To combine these, find a common denominator:
[tex]\[
-\frac{1}{6} x = -\frac{5}{30} x
\][/tex]
[tex]\[
\frac{1}{5} x = \frac{6}{30} x
\][/tex]
Combining these:
[tex]\[
-\frac{5}{30} x + \frac{6}{30} x = \frac{1}{30} x
\][/tex]
So, the trinomial is:
[tex]\[
\frac{2}{15} x^2 + \frac{1}{30} x - \frac{1}{4}
\][/tex]
Step 5: Converting to decimal form for simplicity:
[tex]\[
\frac{2}{15} x^2 = 0.133333333333333 x^2
\][/tex]
[tex]\[
\frac{1}{30} x = 0.0333333333333334 x
\][/tex]
[tex]\[
-\frac{1}{4} = -0.25
\][/tex]
Bringing it all together, the trinomial in simplest form is:
[tex]\[
0.133333333333333 x^2 + 0.0333333333333334 x - 0.25
\][/tex]
This is the product expressed as a trinomial in its simplest form.
