To find the product of [tex]\(\left(\frac{1}{2} x^3\right)(-10 x)\left(\frac{1}{5} x^2\right)\)[/tex] and verify the result for [tex]\(x = 1\)[/tex], follow these steps:
### Step 1: Write down the expression
The given expression is:
[tex]\[
\left(\frac{1}{2} x^3\right)(-10 x)\left(\frac{1}{5} x^2\right)
\][/tex]
### Step 2: Multiply the coefficients
First, identify and multiply the numerical coefficients:
- Coefficient of the first term: [tex]\(\frac{1}{2}\)[/tex]
- Coefficient of the second term: [tex]\(-10\)[/tex]
- Coefficient of the third term: [tex]\(\frac{1}{5}\)[/tex]
Multiplying these together:
[tex]\[
\left(\frac{1}{2}\right) \times (-10) \times \left(\frac{1}{5}\right) = \frac{1}{2} \times (-10) \times \frac{1}{5} = -1
\][/tex]
### Step 3: Multiply the [tex]\(x\)[/tex] terms
Now, multiply the [tex]\(x\)[/tex] terms together:
- [tex]\(x^3\)[/tex] from the first term
- [tex]\(x\)[/tex] from the second term
- [tex]\(x^2\)[/tex] from the third term
Combine these exponents (since [tex]\(x^a \times x^b = x^{a+b}\)[/tex]):
[tex]\[
x^3 \times x \times x^2 = x^{3+1+2} = x^6
\][/tex]
### Step 4: Combine coefficients and [tex]\(x\)[/tex] terms
Combining the coefficient and the [tex]\(x\)[/tex] terms, we get:
[tex]\[
-1 \times x^6 = -x^6
\][/tex]
Thus, the product of [tex]\(\left(\frac{1}{2} x^3\right)(-10 x)\left(\frac{1}{5} x^2\right)\)[/tex] simplifies to:
[tex]\[
-\boldsymbol{x^6}
\][/tex]
### Step 5: Verify the result for [tex]\(x = 1\)[/tex]
To verify the result for [tex]\(x = 1\)[/tex], substitute [tex]\(x = 1\)[/tex] into the simplified expression [tex]\(-x^6\)[/tex]:
[tex]\[
-x^6 = -1^6 = -1
\][/tex]
Therefore, when [tex]\(x = 1\)[/tex], the expression evaluates to:
[tex]\[
\boldsymbol{-1}
\][/tex]
### Summary
Thus, the product of [tex]\(\left(\frac{1}{2} x^3\right)(-10 x)\left(\frac{1}{5} x^2\right)\)[/tex] is [tex]\(-x^6\)[/tex], and when [tex]\(x = 1\)[/tex], the result is [tex]\(-1\)[/tex].
