Problem 10 - Precalculus : Building Concepts and Connections

Question

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$(x)=x^{3}+2 x^{2}-2 x-4 ; x=\sqrt{2},-\sqrt{3}$$

Step-by-Step Solution

Answer

The number \(\sqrt{2}\) is a zero of the polynomial, but \(-\sqrt{3}\) is not.

Step 1: Substituting \(\sqrt{2}\) into the polynomial

Plugging \(\sqrt{2}\) in place of 'x' in the polynomial gives:\[(\sqrt{2})^{3}+2 (\sqrt{2})^{2}-2 \sqrt{2}-4 = 2\sqrt{2} + 4 - 2\sqrt{2} -4 = 0\]Therefore \(\sqrt{2}\) is a zero of the polynomial.

Step 2: Substituting \(-\sqrt{3}\) into the polynomial

Next, substitute \(-\sqrt{3}\) for 'x' in the polynomial to get:\[(-\sqrt{3})^{3}+2 (-\sqrt{3})^{2}-2 (-\sqrt{3})-4 = -3\sqrt{3} + 6 + 2\sqrt{3} - 4 \neq 0\]Hence, \(-\sqrt{3}\) is not a zero of the polynomial.

Key Concepts

Evaluating Polynomials

When we talk about evaluating polynomials, we are referring to the process of calculating the value of a polynomial function for a specific input. Polynomials are algebraic expressions that involve sums of powers of a variable. For example, in the expression \(x^3 + 2x^2 - 2x - 4\), the variable is \(x\), and it has different powers or exponents, specifically 3, 2, and 1.

To evaluate a polynomial at a given number, we replace every instance of the variable \(x\) with that number. This process allows us to find out the corresponding value of the polynomial for that specific input.
When evaluating, each term of the polynomial—which is the product of the coefficient and the power of \(x\)—has to be computed separately and then all terms are summed up.
Here’s a step-by-step:

• Substitute the given number into the polynomial in place of \(x\).
• Compute each power of \(x\) for the substituted number.
• Multiply each result by its corresponding coefficient.
• Sum all these values to obtain the evaluated result.

It’s an essential step in verifying if a specific value, like \(\sqrt{2}\) or \(-\sqrt{3}\), is a zero of a polynomial.

Zeroes of a Function

The zeros of a function are the input values that make the function's output equal to zero. In simpler terms, they are the solutions to the equation when the function, whether it be a polynomial or another type, equals zero.

For polynomial functions like \(f(x) = x^3 + 2x^2 - 2x - 4\), zeros represent the \(x\)-values where the graph of the polynomial crosses or touches the x-axis. Finding zeros is crucial because it provides insights into the behavior and characteristics of functions, such as their graphs.
To determine if a number is a zero of a given polynomial:

• Evaluate the polynomial at the given number.
• If the result is zero, then the number is indeed a zero of the polynomial.
• If not, then the number is not a zero of the polynomial.

In our case, after evaluating the polynomial, we found that \(\sqrt{2}\) turned the polynomial into zero, while \(-\sqrt{3}\) did not.

Rational Root Theorem

The Rational Root Theorem is a useful tool when dealing with polynomial equations. It provides a way to find possible rational roots, or zeros, of a polynomial with integer coefficients.

The theorem states that if a polynomial has a rational solution, or zero, it is given by \( \frac{p}{q} \), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
To apply the Rational Root Theorem:

• List all factors of the constant term (in \(x^3 + 2x^2 - 2x - 4\), the constant term is \(-4\)).
• List all factors of the leading coefficient (here, it's the coefficient of \(x^3\), which is 1).
• Form all possible fractions of \(\frac{p}{q}\) to find the candidates for rational zeros.
• Evaluate the polynomial at these candidates to see if any give a zero result.

Though not always successful in providing all zeros, it significantly narrows down the possibilities, especially when dealing with polynomials with integer coefficients.