Question

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=2 x^{3}-4 x^{2}-10 x+20 $$

Short answer

The real zeros are 2, \(\sqrt{5}\), and \(-\sqrt{5}\). The polynomial factors as \(2(x - 2)(x - \sqrt{5})(x + \sqrt{5})\).

Step-by-step solution

- Identify Potential Rational Zeros

Use the Rational Zeros Theorem to list all potential rational zeros. The Rational Zeros Theorem states that the possible rational zeros of a polynomial function are the fractions formed by the factors of the constant term divided by the factors of the leading coefficient. For the polynomial \(f(x) = 2x^3 - 4x^2 - 10x + 20\), the constant term is 20 and the leading coefficient is 2. The factors of 20 are ±1, ±2, ±4, ±5, ±10, ±20, and the factors of 2 are ±1, ±2. Therefore, the potential rational zeros are: \[ ±1, ±2, ±4, ±5, ±10, ±20, ±\frac{1}{2}, ±\frac{5}{2}, ±\frac{10}{2} = ±1, ±2, ±5. \]

- Test Potential Zeros Using Synthetic Division

Test each potential rational zero using synthetic division to determine if it is a zero of the polynomial. Start with 2: - For \(x = 2\): The synthetic division of \(f(x)\) by \(x-2\) yields: \[\begin{array}{r|rrrr}2 & 2 & -4 & -10 & 20 \ & & 4 & 0 & -20 \hline & 2 & 0 & -10 & 0 \end{array}\]The remainder is 0, so 2 is a zero.

- Factor Polynomial by Found Zero

Since \(x = 2\) is a zero, factor \(x - 2\) from the polynomial. The quotient obtained from synthetic division is \(2x^2 - 10\). Check if this quadratic can be factored further.

- Factor the Quadratic

Factor the quadratic \(2x^2 - 10\). First factor out the common factor of 2 to get \(2(x^2 - 5)\). Recognize that \(x^2 - 5\) can be written as \((x + \sqrt{5})(x - \sqrt{5})\). Thus, the factorization of the quadratic is: \[2(x - \sqrt{5})(x + \sqrt{5})\]

- Write the Complete Factorization

Substitute back the original factor \(x-2\) to get the complete factorization of the polynomial: \[ 2(x - 2)(x - \sqrt{5})(x + \sqrt{5}) \]

- List All Real Zeros

The real zeros of the polynomial \(f(x) = 2x^3 - 4x^2 - 10x + 20\) are: 2, \(\sqrt{5}\), and \(-\sqrt{5}\).

Key concepts

synthetic division

Synthetic division is a shortcut method to divide polynomials, which is especially handy for evaluating polynomials at given values and testing potential zeros. It simplifies the process and avoids the more complex long division.
Here’s how synthetic division works:

• Write down the coefficients of the polynomial in order.
• Choose a potential zero to test (usually a value you believe might zero out the polynomial).
• Bring down the leading coefficient without changing it.
• Multiply the potential zero by the number just brought down and put the result under the next coefficient.
• Add the column of numbers and write the result underneath, then repeat the multiply-add process until finished.

For example, in our exercise, synthetic division was used to test if 2 is a zero for the polynomial function. The process revealed that 2 is indeed a zero since the remainder was 0. This helped us to break down the polynomial further in the factorization process.

polynomial factorization

Polynomial factorization involves expressing a polynomial as the product of its factors. This is crucial for solving polynomial equations, analyzing their properties, and finding zeros.
After identifying a zero using the Rational Zeros Theorem and synthetic division, the polynomial can be broken down into simpler quadratic or linear factors.
In the example exercise, once we determined that 2 is a zero, we factored out \(x - 2\), which left us with a quadratic polynomial \(2x^2 - 10\).
Next, we factored the quadratic polynomial further. Factoring out the common number 2, we get \(2(x^2 - 5)\). Recognizing that \(x^2 - 5\) is a difference of squares, we wrote it as \((x + \sqrt{5})(x - \sqrt{5}))\). Thus, the full factorization of the original polynomial is \(2(x - 2)(x - \sqrt{5})(x + \sqrt{5}))\).

real zeros

Real zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. Mathematically, they are the roots or solutions of the polynomial equation. Finding these zeros is essential in graphing polynomials and understanding their behavior.
The Rational Zeros Theorem provides a way to find possible rational zeros, which are candidates for the actual zeros. After listing the potential zeros, we use synthetic division to verify which of these are actual zeros.
In our exercise, synthetic division helped us confirm that \(2\) is a real zero. We then factored the polynomial further and identified additional zeros from the factored form. After completely factoring the polynomial as \(2(x - 2)(x - \sqrt{5})(x + \sqrt{5}))\), we identified the real zeros as \(2\), \(\sqrt{5}\) and \(-\sqrt{5}\).\
Understanding real zeros is important because:

• They show where the polynomial graph intersects the x-axis.
• They help in solving polynomial equations.
• They provide insights into the behavior of polynomial functions over defined intervals.