Question

Is \(\frac{3}{5}\) a zero of \(f(x)=2 x^{6}-5 x^{4}+x^{3}-x+1 ?\) Explain.

Short answer

\( \frac{3}{5} eq 0 \) for \( f(x) = 2x^{6} - 5x^{4} + x^{3} - x + 1 \), so \( \frac{3}{5} \) is not a zero.

Step-by-step solution

- Understand the Problem

To determine if \(\frac{3}{5}\) is a zero of the function \(f(x)=2 x^{6}-5 x^{4}+x^{3}-x+1\), substitute \(x = \frac{3}{5}\) into the function and see if the result is zero.

- Substitute \(x = \frac{3}{5}\)

Substitute \(x = \frac{3}{5}\) into the function \(f(x)\): \(f\bigg(\frac{3}{5}\bigg) = 2\bigg(\frac{3}{5}\bigg)^{6} - 5\bigg(\frac{3}{5}\bigg)^{4} + \bigg(\frac{3}{5}\bigg)^{3} - \bigg(\frac{3}{5}\bigg) + 1\).

- Calculate \(\bigg(\frac{3}{5}\bigg)^{6}\)

Calculate \(\bigg(\frac{3}{5}\bigg)^{6}\): \( \bigg(\frac{3}{5}\bigg)^{6} = \bigg(\frac{3^6}{5^6}\bigg) = \bigg(\frac{729}{15625}\bigg)\).

- Calculate \(\bigg(\frac{3}{5}\bigg)^{4}\)

Calculate \(\bigg(\frac{3}{5}\bigg)^{4}\): \( \bigg(\frac{3}{5}\bigg)^{4} = \bigg(\frac{3^4}{5^4}\bigg) = \bigg(\frac{81}{625}\bigg)\).

- Calculate \(\bigg(\frac{3}{5}\bigg)^{3}\)

Calculate \(\bigg(\frac{3}{5}\bigg)^{3}\): \( \bigg(\frac{3}{5}\bigg)^{3} = \bigg(\frac{3^3}{5^3}\bigg) = \bigg(\frac{27}{125}\bigg)\).

- Calculate \(\bigg(\frac{3}{5}\bigg)\)

Calculate \(\bigg(\frac{3}{5}\bigg)\): \( \bigg(\frac{3}{5}\bigg)\).

- Compute \(f\bigg(\frac{3}{5}\bigg)\)

Now, compute \(f\bigg(\frac{3}{5}\bigg)\): \(f\bigg(\frac{3}{5}\bigg) = 2 \bigg(\frac{729}{15625}\bigg) - 5 \bigg(\frac{81}{625}\bigg) + \bigg(\frac{27}{125}\bigg) - \bigg(\frac{3}{5}\bigg) + 1\).

- Simplify the Expression

Simplify each term, and then combine them: \(2 \bigg(\frac{729}{15625}\bigg) - 5 \bigg(\frac{81}{625}\bigg) + \bigg(\frac{27}{125}\bigg) - \bigg(\frac{3}{5}\bigg) + 1\).

- Conclusion

If the expression equals zero, then \( \frac{3}{5} \) is a zero of the polynomial. Otherwise, it is not.

Key concepts

Zeros of a Polynomial

Identifying the **zeros of a polynomial** involves finding the values of the variable that make the function equal to zero. For a given polynomial function, such as \(f(x) = 2x^6 - 5x^4 + x^3 - x + 1\), we need to determine if a particular value of \(x\), like \(\frac{3}{5}\), satisfies the condition \(f(x) = 0\).

To check this, we substitute \(x = \frac{3}{5}\) into the polynomial. If the resulting expression equals zero, then \(\frac{3}{5}\) is indeed a zero of the function. Finding zeros helps us understand the roots and the behavior of polynomial functions.

Function Substitution

To determine if \(\frac{3}{5}\) is a zero of a polynomial function like \(f(x) = 2x^6 - 5x^4 + x^3 - x + 1\), we use **function substitution**. This means we replace \(x\) in the polynomial with \(\frac{3}{5}\) and calculate the result.

Let's substitute the value:
\(f\left(\frac{3}{5}\right) = 2\left(\frac{3}{5}\right)^6 - 5\left(\frac{3}{5}\right)^4 + \left(\frac{3}{5}\right)^3 - \left(\frac{3}{5}\right) + 1\).

Conducting this substitution helps us transform the function into a numerical form that we can easily evaluate.

Polynomial Evaluation

After we have substituted the value into the polynomial, the next step is **polynomial evaluation**. This involves calculating the specific value of each term.

For example:

• \(\left(\frac{3}{5}\right)^6 = \frac{729}{15625}\)
• \(\left(\frac{3}{5}\right)^4 = \frac{81}{625}\)
• \(\left(\frac{3}{5}\right)^3 = \frac{27}{125}\)
• \(\left(\frac{3}{5}\right) = \frac{3}{5}\)

We substitute these results back into the expression:
\(2 \left(\frac{729}{15625}\right) - 5 \left(\frac{81}{625}\right) + \left(\frac{27}{125}\right) - \left(\frac{3}{5}\right) + 1\).

Finally, we simplify the expression by combining like terms and reducing fractions as needed.

Rational Exponents

In polynomial functions, **rational exponents** often appear. These exponents can represent fractions or whole numbers. In our problem, we encounter rational exponents such as: \(\left(\frac{3}{5}\right)^6\) or \(\left(\frac{3}{5}\right)^4\).

Rational exponents follow specific rules:

• \(a^{m/n} = \sqrt[n]{a^m} \)
• \(a^{m/n} = (a^{1/n})^m\)

Using these rules, it's easier to understand and compute the terms in the polynomial. For instance, \(\left(\frac{3}{5}\right)^6\) simplifies to \(\frac{3^6}{5^6}\), making the computation straightforward. Understanding rational exponents is crucial for evaluating and simplifying polynomial functions.