Problem 34 - College Algebra

Question

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(x^{4}+2 x^{3}-9 x^{2}-2 x+8=0\)

Step-by-Step Solution

Answer

Real zeros are 1, -1, 2, and -4.

Step 1: Write Down the Polynomial

The given polynomial is \(x^{4} + 2x^{3} - 9x^{2} - 2x + 8 = 0\). This is a fourth-degree polynomial.

Step 2: Identify the Constant and Leading Coefficient

The constant term of the polynomial is 8, and the leading coefficient is 1.

Step 3: Find the Possible Rational Zeros

According to the Rational Zero Theorem, possible rational zeros are of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term (8) and \(q\) is a factor of the leading coefficient (1). Factors of 8 are \(\pm 1, \pm 2, \pm 4, \pm 8\), and factors of 1 are \(\pm 1\). Thus, possible rational zeros are \(\pm 1, \pm 2, \pm 4, \pm 8\).

Step 4: Test Possible Zeros Using Synthetic Division

Use synthetic division to test each possible zero: 1, 2, 4, and then their negatives, to find which value(s) yield a remainder of zero when dividing the polynomial.

Step 5: Find a Zero Successful Test

After performing synthetic division, test shows that \(x = 1\) is a zero, as the remainder is zero.

Step 6: Factor the Polynomial Using Found Zeros

With \(x = 1\) confirmed as a zero, the polynomial can be factored as \((x - 1)(x^{3} + 3x^{2} - 6x - 8) = 0\).

Step 7: Solve the Cubic Polynomial

Next, solve the cubic polynomial \(x^{3} + 3x^{2} - 6x - 8 = 0\) using synthetic division again or by finding another rational zero. Testing shows \(x = -1\) as another zero.

Step 8: Further Factor and Solve

Factor out \(x = -1\) from \(x^{3} + 3x^{2} - 6x - 8\), to get \((x + 1)(x^{2} + 2x - 8)\).

Step 9: Solve the Quadratic

Solve the quadratic equation \(x^{2} + 2x - 8 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 2\), \(c = -8\).

Step 10: Determining the Quadratic Solutions

Calculating yields \(x = 2\) and \(x = -4\). The full factorization is \((x - 1)(x + 1)(x - 2)(x + 4) = 0\).

Step 11: List All Real Zeros

The real zeros of the polynomial are \(x = 1, -1, 2,\) and \(-4\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \((x - c)\). It is more efficient than long division and is especially useful when applying the Rational Zero Theorem to find potential zeros of a polynomial. Here's how you do synthetic division:

• First, bring down the leading coefficient of the polynomial.
• Next, multiply this number by the possible zero you are testing and write the result under the next coefficient. Add these two numbers together.
• Continue this process for each coefficient in the polynomial.
• Finally, the last number you have left is the remainder. If it is zero, the test zero is a valid zero of the polynomial.

For the polynomial given: \(x^4 + 2x^3 - 9x^2 - 2x + 8\), we used synthetic division to test potential zeros like \(x = 1\) and \(x = -1\). When the remainder is zero, it indicates the test zero divides the polynomial exactly, confirming it's a real zero.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler, multiplyable factors, which multiply together to yield the original polynomial. This process often follows the discovery of a polynomial’s zeros, using techniques like synthetic division.

• Start by identifying one or more zeros using methods like the Rational Zero Theorem.
• Once a zero is found, factor it out of the polynomial. Each factor corresponds to a zero and simplifies the polynomial.
• Continue factoring until you reach simpler forms, such as linear or quadratic expressions.

In the solution given, we started by factoring out \(x = 1\) from the original polynomial. We then factored \(x = -1\) out from the resulting cubic expression. This simplification led us to a quadratic polynomial that could be solved further through different methods like the quadratic formula.

Quadratic Formula

In the context of solving polynomial equations, the quadratic formula is a reliable tool for finding zeros of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here’s how it works:

• Identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute them into the formula.
• Calculate the discriminant, \(b^2 - 4ac\). If it’s positive, there are two real solutions. Zero yields one real solution, and negative provides complex solutions.

For the equation \(x^2 + 2x - 8 = 0\), \(a = 1\), \(b = 2\), and \(c = -8\). Plugging these values into the quadratic formula gives solutions \(x = 2\) and \(x = -4\), which are the remaining zeros of the original polynomial. This step completes the factorization process by accessing all real solutions.