Kari is incorrect in her conclusion that [tex]\(x^2 + x\)[/tex] is equivalent to [tex]\(x(2x + 1)\)[/tex] simply because the two expressions have the same value when [tex]\(x = 0\)[/tex]. To determine if two expressions are equivalent, they must yield the same value for all values of [tex]\(x\)[/tex], not just a single value.
Let's examine the given expressions step-by-step for [tex]\(x = 0\)[/tex] and other values.
1. Evaluate [tex]\(x^2 + x\)[/tex] when [tex]\(x = 0\)[/tex]:
[tex]\[
x^2 + x = 0^2 + 0 = 0
\][/tex]
2. Evaluate [tex]\(x(2x + 1)\)[/tex] when [tex]\(x = 0\)[/tex]:
[tex]\[
x(2x + 1) = 0(2 \cdot 0 + 1) = 0 \cdot 1 = 0
\][/tex]
Indeed, both expressions have the same value of 0 when [tex]\(x = 0\)[/tex].
3. Check if the expressions are equivalent for other values of [tex]\(x\)[/tex]:
Let's choose another value, say [tex]\(x = 1\)[/tex]:
- For [tex]\(x^2 + x\)[/tex]:
[tex]\[
x^2 + x = 1^2 + 1 = 1 + 1 = 2
\][/tex]
- For [tex]\(x(2x + 1)\)[/tex]:
[tex]\[
x(2x + 1) = 1(2 \cdot 1 + 1) = 1(2 + 1) = 1 \cdot 3 = 3
\][/tex]
When [tex]\(x = 1\)[/tex], the expression [tex]\(x^2 + x\)[/tex] evaluates to 2, while [tex]\(x(2x + 1)\)[/tex] evaluates to 3.
4. Compare the structures of the expressions:
We can further verify by expanding and simplifying both expressions generally:
- [tex]\(x^2 + x\)[/tex] is already in its simplest form.
- Expanding [tex]\(x(2x + 1)\)[/tex]:
[tex]\[
x(2x + 1) = 2x^2 + x
\][/tex]
We observe that [tex]\(x^2 + x\)[/tex] simplifies to [tex]\(x^2 + x\)[/tex], while [tex]\(x(2x + 1)\)[/tex] simplifies to [tex]\(2x^2 + x\)[/tex]. Clearly, [tex]\(x^2 + x\)[/tex] and [tex]\(2x^2 + x\)[/tex] are not identical expressions.
Hence, [tex]\(x^2 + x\)[/tex] and [tex]\(x(2x + 1)\)[/tex] are not equivalent expressions overall, even though they give the same value for [tex]\(x = 0\)[/tex].
Therefore, Kari is incorrect because two expressions that are equivalent must have the same value for all values of [tex]\(x\)[/tex], not just for a single value.
