Problem 21 - Algebra 2

Question

Find all of the rational zeros of each function. \(p(x)=x^{4}+10 x^{3}+33 x^{2}+38 x+8\)

Step-by-Step Solution

Answer

Rational zeros are \(-1\) and \(+1\).

Step 1: List Possible Rational Zeros

According to the Rational Root Theorem, possible rational zeros of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. Here, the constant term is 8 and the leading coefficient is 1.Possible factors of 8 are: \(\pm 1, \pm 2, \pm 4, \pm 8\)Since the leading coefficient is 1, the possible rational zeros are the factors themselves: \(\pm 1, \pm 2, \pm 4, \pm 8\).

Step 2: Test Possible Zeros using Synthetic Division

We will test each potential rational zero by performing synthetic division until we find a zero remainder.**Testing \(+1\):** The remainder is not zero.**Testing \(-1\):** The remainder is zero. So, \(-1\) is a root. We now have a quotient \(x^3 + 9x^2 + 24x + 8\).We will continue the synthetic division process for this reduced polynomial.

Step 3: Find Remaining Zeros of Reduced Polynomial

Perform synthetic division on the reduced polynomial \(x^3 + 9x^2 + 24x + 8\) found from previous steps.**Testing \(+1\):** Test succeeds, remainder is zero. The quotient becomes \(x^2 + 8x + 8\).**Testing \(-1\):** No need, because \(+1\) gave a zero already in this round.Continue finding zeros for \(x^2 + 8x + 8\).

Step 4: Solve Quadratic Equation

The reduced polynomial is now \(x^2 + 8x + 8\). We will find its zeros using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 1\), \(b = 8\), and \(c = 8\).Calculate the discriminant: \(b^2 - 4ac = 64 - 32 = 32\).Thus, the roots are:\[x = \frac{-8 \pm \sqrt{32}}{2} = \frac{-8 \pm 4\sqrt{2}}{2} = -4 \pm 2\sqrt{2}\].

Step 5: Compile All Rational Zeros

From synthetic division, we found the rational zeros \(-1\) and \(+1\). The remaining roots \(-4 \pm 2\sqrt{2}\) are not rational. Therefore, the rational zeros of the polynomial are \(-1\) and \(+1\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a vital tool for finding rational solutions to polynomial equations. When solving a polynomial like \(p(x) = x^4 + 10x^3 + 33x^2 + 38x + 8\), this theorem helps us identify possible rational roots by examining the factors of the constant term and the leading coefficient.

Here's how it works:

• The constant term is 8, and its factors are \(\pm 1, \pm 2, \pm 4, \pm 8\).
• The leading coefficient is 1, whose factors are just \(\pm 1\).

This means any potential rational root of the polynomial must be a factor of 8. Thus, the possible rational roots are \(\pm 1, \pm 2, \pm 4, \pm 8\). This theorem simplifies the search for roots and helps us focus on only these feasible options.

Synthetic Division

Once we have a list of possible rational roots, synthetic division becomes very handy. It helps us test each potential zero quickly. Synthetic division provides a streamlined method to divide polynomials, particularly useful here to check if a number is indeed a root.

Let's say we try \(-1\) as a possible root of our polynomial. Performing synthetic division on \(x^4 + 10x^3 + 33x^2 + 38x + 8\) with \(-1\), we find the remainder is zero. This result confirms \(-1\) is a root. The process is as follows:

• Line up the coefficients: \(1, 10, 33, 38, 8\).
• Write \(-1\) outside the synthetic division box.
• Bring down the first coefficient. Multiply and add successively through the process.
• If the last number (remainder) is zero, \(-1\) is indeed a root.

With \(-1\) confirmed as a root, we simplify our polynomial to \(x^3 + 9x^2 + 24x + 8\) and continue testing for more roots.

Polynomial Roots

Finding all the roots of a polynomial, including rational and irrational ones, involves breaking down the polynomial into factors and checking the reduced forms. After synthetic division, polynomial \(x^3 + 9x^2 + 24x + 8\) is tested for further roots.

We successfully divide by \(+1\), leaving a quotient of \(x^2 + 8x + 8\). We are not only interested in rational zeros (such as \(-1\) and \(+1\)), but we also look to find all the potential roots of the reduced polynomial. Sometimes these roots can be irrational, requiring additional methods like the quadratic formula.

In summary, synthetic division assists in isolating rational roots, which then helps concentrate further efforts on the remaining equations.

Quadratic Formula

To address the quadratic part of our polynomial \(x^2 + 8x + 8\), the quadratic formula becomes immensely useful. This powerful formula calculates the roots of any quadratic equation and handles cases where roots are not conveniently rational. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For \(x^2 + 8x + 8\), we plug in:

• \(a = 1\)
• \(b = 8\)
• \(c = 8\)

Calculating the discriminant: \(b^2 - 4ac = 64 - 32 = 32\). This is positive, indicating two distinct roots exist. Solving with the formula gives:\[x = \frac{-8 \pm \sqrt{32}}{2} = -4 \pm 2\sqrt{2}\].These solutions include irrational numbers, showing the critical role of the quadratic formula in revealing all possible solutions beyond just the rational ones.