Determine if the statement is true or false. If a statement is false, explain why. Suppose that $$ f(x) $$ is a polynomial, and that $$ a $$ and $$ b $$ are real numbers where $$ a

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The statement is **True**.

To understand why, let's analyze the conditions given:

1. **Polynomial Behavior**: A polynomial function $$ f(x) $$ is continuous everywhere on the real line. This means that as $$ x $$ moves from $$ a $$ to $$ b $$, the value of $$ f(x) $$ will change continuously.

2. **Values at Endpoints**: The statement specifies that $$ f(a) < 0 $$ and $$ f(b) < 0 $$. This means that at both endpoints of the interval $$ [a, b] $$, the polynomial takes negative values.

3. **Intermediate Value Theorem**: According to the Intermediate Value Theorem, if a continuous function takes on two different values at two points, it must take on every value in between those two points at least once. However, since both $$ f(a) $$ and $$ f(b) $$ are negative, there is no value of $$ f(x) $$ that can be zero (or positive) between $$ a $$ and $$ b $$.

4. **Conclusion**: Since $$ f(x) $$ does not cross the x-axis (i.e., does not equal zero) anywhere in the interval $$ [a, b] $$, we conclude that $$ f(x) $$ has no real zeros in that interval.

Thus, the statement is true: if $$ f(a) < 0 $$ and $$ f(b) < 0 $$, then $$ f(x) $$ has no real zeros on the interval $$ [a, b] $$.
The statement is **True**. If $$ f(a) < 0 $$ and $$ f(b) < 0 $$ for a continuous polynomial $$ f(x) $$ on the interval $$ [a, b] $$, then $$ f(x) $$ does not cross the x-axis within that interval, meaning it has no real zeros there.

Determine if the statement is true or false. If a statement is false, explain why. Suppose that \( f(x) \) is a polynomial, and that \( a \) and \( b \) are real numbers where \( a

Ask by Powell Clarke. in the United States

Mar 16,2025

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Determine if the statement is true or false. If a statement is false, explain why. Suppose that \( f(x) \) is a polynomial, and that \( a \) and \( b \) are real numbers where \( a Ask by Powell Clarke. in the United States Mar 16,2025