Question

Is \(\frac{2}{3}\) a zero of \(f(x)=x^{7}+6 x^{5}-x^{4}+x+2 ?\) Explain.

Short answer

No, \( \frac{2}{3} \) is not a zero of the function \( f(x) = x^{7} + 6x^{5} - x^{4} + x + 2 \).

Step-by-step solution

- Understand the problem

Determine whether \( \frac{2}{3} \) is a zero of the function \( f(x) = x^{7} + 6x^{5} - x^{4} + x + 2 \). This can be done by substituting \( \frac{2}{3} \) for \( x \) in the function and checking if the result is equal to zero.

- Substitute \( \frac{2}{3} \) into the function

Substitute \( x = \frac{2}{3} \) into \( f(x) \). Calculate \( f \left( \frac{2}{3} \right) \): \[ f \left( \frac{2}{3} \right) = \left( \frac{2}{3} \right)^{7} + 6 \left( \frac{2}{3} \right)^{5} - \left( \frac{2}{3} \right)^{4} + \left( \frac{2}{3} \right) + 2 \]

- Calculate each term

Compute each term individually: \[ \left( \frac{2}{3} \right)^{7} = \frac{128}{2187} \] \[ 6 \left( \frac{2}{3} \right)^{5} = 6 \cdot \frac{32}{243} = \frac{192}{243} = \frac{64}{81} \] \[ - \left( \frac{2}{3} \right)^{4} = - \frac{16}{81} \] \[ \left( \frac{2}{3} \right) = \frac{2}{3} \] \[ 2 = 2 \]

- Sum up all terms

Add up the terms: \[ f \left( \frac{2}{3} \right) = \frac{128}{2187} + \frac{64}{81} - \frac{16}{81} + \frac{2}{3} + 2 \]

- Simplify the sum

Convert each term to a common denominator if possible: \[ f \left( \frac{2}{3} \right) = \frac{128}{2187} + \frac{64 - 16}{81} + \frac{2}{3} + 2 \] Simplify: \[ f \left( \frac{2}{3} \right) = \frac{128}{2187} + \frac{48}{81} + \frac{2}{3} + 2 \] \[ f \left( \frac{2}{3} \right) \approx 0.0586 + 0.5926 + 0.6667 + 2 = 3.3179 \]

Conclusion

Since \( f \left( \frac{2}{3} \right) \) is not equal to zero, \( \frac{2}{3} \) is not a zero of the function \( f(x) \).

Key concepts

Polynomial Functions

Polynomial functions are expressions involving a sum of powers in one or more variables multiplied by coefficients. They're usually written in the form: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]where each \( a_i \) represents a coefficient and \( n \) is a non-negative integer. Polynomial functions can have many terms, and their degree is determined by the highest power of the variable. For instance, in the function \( f(x) = x^7 + 6x^5 - x^4 + x + 2 \), the highest power of x is 7, making it a 7th-degree polynomial. Polynomials are central in algebra and calculus because they can model various real-world situations and describe complex relationships between quantities.

Substitution Method

The substitution method involves replacing a variable with a given value to determine the outcome. This technique is particularly useful in verifying potential zeros for polynomial functions. To check if \( \frac{2}{3} \) is a zero of \( f(x) = x^7 + 6x^5 - x^4 + x + 2 \), we substitute \( x = \frac{2}{3} \) into the polynomial. If the resulting value, \( f( \frac{2}{3} ) \), equals zero, \( \frac{2}{3} \) is a zero of the polynomial. Performing these substitutions involves precise calculations and algebraic manipulation, which are foundational skills in higher mathematics.

Algebraic Evaluation

Algebraic evaluation is the process of computing the value of a mathematical expression using algebraic operations. In the context of polynomials, we break down the process into manageable steps. For example, to evaluate \( f( \frac{2}{3} ) \) for \( f(x) = x^7 + 6x^5 - x^4 + x + 2 \), we calculate each term separately:

• First, compute \( ( \frac{2}{3} )^7 = \frac{128}{2187} \).
• Next, calculate \( 6 ( \frac{2}{3} )^5 = 6 \cdot \frac{32}{243} = \frac{64}{81} \).
• Then, \( - ( \frac{2}{3} )^4 = - \frac{16}{81} \).
• Also, \( ( \frac{2}{3} ) = \frac{2}{3} \).
• Lastly, we have \( 2 \).

Adding these values and simplifying the expression will indicate whether the substituted value makes the polynomial equal to zero.

Rational Numbers

A rational number can be expressed as the quotient of two integers, where the denominator is not zero. For example, \( \frac{2}{3} \) is a rational number. Rational numbers are essential in algebra because they can often represent solutions to equations and zeros of polynomial functions. When we evaluate whether a rational number is a zero of a polynomial, we substitute it for the variable and perform arithmetic operations to see if the polynomial equals zero. If, after substitution and simplification, the result is zero, the rational number is a zero of the polynomial. Developing a solid understanding of rational numbers and their properties helps in solving equations and understanding algebraic structures.