Pregunta
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Which of the following is a polynomial? Select all that apply. \( f(x)=3 x\left(x^{4}-6 x^{3}+1\right) \) \( g(x)=5 x^{7}+2 x^{-3}-7 x+10 \) \( y=8 \sqrt{x}+6 x-4 \) \( h(x)=5^{x}+8 x^{3}-2 x \) \( f(x)=-3 x^{4}+\frac{5}{2} x \) \( f \)

Ask by Hills Fernandez. in the United States
Mar 06,2025

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The polynomials are \( f(x) = 3 x (x^{4} - 6 x^{3} + 1) \) and \( f(x) = -3 x^{4} + \frac{5}{2} x \).

Solución

To determine which of the given functions are polynomials, we need to recall the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers and coefficients that are real numbers. Let's analyze each function one by one: 1. **\( f(x) = 3 x (x^{4} - 6 x^{3} + 1) \)**: - This is a product of a constant and a polynomial (since \( x^{4} - 6 x^{3} + 1 \) is a polynomial). - Therefore, \( f(x) \) is a polynomial. 2. **\( g(x) = 5 x^{7} + 2 x^{-3} - 7 x + 10 \)**: - This function contains the term \( 2 x^{-3} \), which has a negative exponent. - Therefore, \( g(x) \) is not a polynomial. 3. **\( y = 8 \sqrt{x} + 6 x - 4 \)**: - The term \( 8 \sqrt{x} \) can be rewritten as \( 8 x^{1/2} \), which has a fractional exponent. - Therefore, \( y \) is not a polynomial. 4. **\( h(x) = 5^{x} + 8 x^{3} - 2 x \)**: - The term \( 5^{x} \) is an exponential function, not a polynomial. - Therefore, \( h(x) \) is not a polynomial. 5. **\( f(x) = -3 x^{4} + \frac{5}{2} x \)**: - This function consists of terms with non-negative integer powers of \( x \) (specifically \( x^{4} \) and \( x^{1} \)). - Therefore, this \( f(x) \) is a polynomial. In summary, the functions that are polynomials are: - \( f(x) = 3 x (x^{4} - 6 x^{3} + 1) \) - \( f(x) = -3 x^{4} + \frac{5}{2} x \) Thus, the polynomials are: - \( f(x) \) (first one) - \( f(x) \) (last one)

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Extra Insights

A polynomial is an expression that involves a sum of powers in one or more variables multiplied by coefficients, without any negative or fractional powers. So, let's dive into the exciting world of polynomials! First, the function \( f(x)=3 x\left(x^{4}-6 x^{3}+1\right) \) is a polynomial! By distributing that, you'll see it can be expressed as \( 3x^{5} - 18x^{4} + 3x \), which consists of whole-number powers of \( x \). On the flip side, \( g(x)=5 x^{7}+2 x^{-3}-7 x+10 \) is not a polynomial because of the \( 2 x^{-3} \) term—negative exponents just don’t belong in our polynomial party! Similarly, \( y=8 \sqrt{x}+6 x-4 \) and \( h(x)=5^{x}+8 x^{3}-2 x \) are off the guest list too due to their square root and exponential components, respectively. Last but not least, \( f(x)=-3 x^{4}+\frac{5}{2} x \) is indeed a polynomial despite that pesky fraction; it just needs whole number powers! So, let’s recap: \( f(x) \) and \( f(x) = -3 x^{4} + \frac{5}{2} x \) are our polynomial stars! 🎉

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