Problem 5 - Algebra 2

Question

Find all of the zeros of each function. \(p(x)=x^{3}+2 x^{2}-3 x+20\)

Step-by-Step Solution

Answer

The zeros are \( x = -5, \sqrt{3}, -\sqrt{3} \).

Step 1: Understand the Problem

We are tasked with finding the zeros of the polynomial function \( p(x) = x^3 + 2x^2 - 3x + 20 \). The zeros of a function are the \( x \)-values for which \( p(x) = 0 \).

Step 2: Use Rational Root Theorem

The Rational Root Theorem suggests that any rational root, \( \frac{p}{q} \), of the polynomial is such that \( p \) divides the constant term (20) and \( q \) divides the leading coefficient (1). Therefore, potential rational roots are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).

Step 3: Test Potential Roots Using Synthetic Division

Try some potential roots to see if they give \( p(x) = 0 \). For instance, perform synthetic division on \( x = -2 \).

Step 4: Synthetic Division for \( x = -2 \)

Using synthetic division for \( x = -2 \): \[\begin{array}{c|cccc}-2 & 1 & 2 & -3 & 20 \ & & -2 & 0 & 6 \\hline & 1 & 0 & -3 & 26 \\end{array}\]The remainder is 26, so \( x = -2 \) is not a root.

Step 5: Continue Testing Other Potential Roots

Test the potential root \( x = -5 \) using synthetic division. The process gives a remainder 0, confirming \( x = -5 \) as a root.

Step 6: Factor the Polynomial

Since \( x = -5 \) is a root, \( (x + 5) \) is a factor. Perform polynomial division to find the quotient. Divide \( x^3 + 2x^2 - 3x + 20 \) by \( x + 5 \), resulting in \( x^2 - 3 \).

Step 7: Solve Quadratic Factor

Solve the remaining quadratic factor: \( x^2 - 3 = 0 \), giving roots \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).

Step 8: Conclusion on Roots

Thus, the roots of \( p(x) = 0 \) are \( x = -5 \), \( x = \sqrt{3} \), and \( x = -\sqrt{3} \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a valuable tool in finding the zeros of polynomial functions. It provides a list of possible rational roots for a polynomial equation. For a polynomial like \( p(x) = x^3 + 2x^2 - 3x + 20 \), the theorem tells us the possible rational roots are in the form \( \frac{p}{q} \). Here, \( p \) must be a factor of the constant term (which is 20 in our polynomial), and \( q \) must be a factor of the leading coefficient (which is 1).

This means potential rational roots for our polynomial are:

• \( \pm 1 \)
• \( \pm 2 \)
• \( \pm 4 \)
• \( \pm 5 \)
• \( \pm 10 \)
• \( \pm 20 \)

This list gives a manageable starting point for testing which of these values could be actual roots. They are not guaranteed to be roots, but they are well worth checking because they satisfy the theorem's conditions.

Synthetic Division

Once we have a list of potential rational roots from the Rational Root Theorem, synthetic division helps us verify these roots quickly. It's a streamlined version of polynomial division, used to determine whether a given potential root actually makes the polynomial equal to zero.

To perform synthetic division:

• Write down the coefficients of the polynomial: for \( x^3 + 2x^2 - 3x + 20 \), the coefficients are \( 1, 2, -3, 20 \).
• Choose a potential root to test, say \( x = -2 \).
• Follow the synthetic division process:
  • Bring down the first coefficient (1) to the bottom row.
  • Multiply it by the potential root (\(-2\)) and add this result to the next coefficient (2), yielding 0.
  • Continue the process to see if the final remainder is 0.

If the final remainder is 0, then the potential root is an actual root of the polynomial. In the case of our polynomial, using synthetic division showed that \( x = -5 \) is a real root because the remainder was 0.

Quadratic Factorization

After confirming a root like \( x = -5 \) through synthetic division, it's possible to factor the polynomial further. With one linear factor identified \((x + 5)\), the next step involves finding the quadratic polynomial that remains.

In our example, dividing \( x^3 + 2x^2 - 3x + 20 \) by \( x + 5 \) yields the quadratic \( x^2 - 3 \). Here, the polynomial splits from a cubic function into the product of a linear function and a quadratic function that needs solving.

The remaining quadratic polynomial can be tackled using various strategies like factoring, completing the square, or using the quadratic formula, depending on its complexity and structure. In this case, \( x^2 - 3 = 0 \) is simple and straightforward to handle by solving for \( x \).

Polynomial Division

Polynomial division is crucial in transitioning from one polynomial form to another and is especially important after discovering a root. Once a root is confirmed, like \( x = -5 \), dividing the original polynomial function by the corresponding factor \((x + 5)\), reduces its degree.

Polynomial division works much like long division with numbers. It simplifies the expression by effectively rehearsing the multiplication in reverse:

• Divide the first term of the dividend by the first term of the divisor, writing the result as the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient and subtract the result from the original polynomial.
• Repeat this process with the new polynomial.

By following this approach for \( x^3 + 2x^2 - 3x + 20 \) divided by \( x + 5 \), the result is the quadratic \( x^2 - 3 \). Solving this lower-degree polynomial equation completes finding all the zeros of the original function.