Problem 4 - Algebra 2

Question

Find all of the rational zeros of each function. \(f(x)=x^{3}-x^{2}-34 x-56\)

Step-by-Step Solution

Answer

The rational zeros of the polynomial are \(-2, 7,\) and \(-4\).

Step 1: Identify the Possible Rational Zeros

According to the Rational Root Theorem, any rational zero of the polynomial \(f(x)\) must be a factor of the constant term divided by a factor of the leading coefficient. For \(f(x) = x^3 - x^2 - 34x - 56\), the constant term is -56 and the leading coefficient is 1. Thus, the possible rational zeros are the factors of -56, which are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\).

Step 2: Test Possible Zeros Using Synthetic Division

Start testing potential zeros using synthetic division. For each candidate zero, divide the polynomial with that candidate and check if the remainder is zero. Begin with \(x = 1\): The remainder is not zero. Continue this process with \(-1, 2, -2, \ldots\) until you find a candidate that results in a zero remainder.

Step 3: Verify Zero and Find Reduced Polynomial

Check through synthetic division that \(x = -2\) results in a zero remainder. The division provides \((x + 2)(x^2 - 3x - 28)\) as the factorization of the polynomial.

Step 4: Factor the Quadratic Polynomial

Factor the quadratic \(x^2 - 3x - 28\) to find the other rational roots. Solving \(x^2 - 3x - 28 = 0\), we factor it as \((x - 7)(x + 4)\).

Step 5: Collect All Rational Zeros

From the factorization, the rational zeros are \(x = -2, 7, -4\). Therefore, all rational zeros for \(f(x)\) are \(-2, 7,\) and \(-4\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method used to divide polynomials when the divisor is of the form \(x - c\). It offers a quicker way to test potential zeros found by the Rational Root Theorem. The process involves setting up the coefficients of the polynomial in a row and iteratively carrying out arithmetic operations to determine the quotient and the remainder. If the remainder is zero, it confirms that \(c\) is a root of the polynomial.

• Line up the coefficients of the polynomial: For \(f(x) = x^3 - x^2 - 34x - 56\), list the coefficients: \([1, -1, -34, -56]\).
• Use the candidate zero, like \(x = -2\), and perform synthetic division.
• Bring down the leading coefficient (1), multiply by \(-2\) (which is \(x\)), and add this to the next coefficient.
• Continue this process across all coefficients.
• If the final number (remainder) is zero, \(x = -2\) is a root.

By performing synthetic division quickly, you can identify whether potential rational zeros are actual zeros without going through long polynomial division steps.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give you the original polynomial. This is particularly useful in solving polynomial equations and finding zeros. In the exercise with \(f(x) = x^3 - x^2 - 34x - 56\), once a zero is found through synthetic division, the polynomial is further factorized into smaller parts.Here:

• Upon identifying \(x = -2\) as a zero, the polynomial is rewritten as \((x + 2)(x^2 - 3x - 28)\).
• The quadratic \(x^2 - 3x - 28\) is factorized further to find more roots.
• This results in \((x - 7)(x + 4)\), providing two additional factors.

Factorization allows us to rewrite polynomials in such a way that makes solving equations and identifying zeros much simpler, by handling each lower-degree factor separately.

Quadratic Equation

A quadratic equation is an equation of the second degree, usually in the format \(ax^2 + bx + c = 0\). Solving these equations is key in polynomial factorization after we break down a larger polynomial and identify one factor. The quadratic \(x^2 - 3x - 28\), derived from our original polynomial, needs to be resolved into its simplest roots.To solve a quadratic equation:

• One common method is factorization, which involves expressing the quadratic in terms of two binomials.
• For \(x^2 - 3x - 28\), it factors into \((x - 7)(x + 4)\).
• This reveals that \(x = 7\) and \(x = -4\) are the solutions (zeros of the equation).

Finding the roots of a quadratic provides valuable insights into the overall solutions of polynomial equations, helping verify the full set of rational zeros.

Rational Zeros

Rational zeros are potential solutions to polynomial equations that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers. The Rational Root Theorem is a useful tool to identify these potential zeros:The theorem suggests:

• Potential rational zeros will be \(\pm\) the factors of the constant term divided by the factors of the leading coefficient.
• For \(f(x) = x^3 - x^2 - 34x - 56\), the constant is \(-56\) and the leading coefficient is 1, meaning possible zeros are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\).

Once a list of potential rational zeros is prepared, testing these through synthetic division helps find the actual rational zeros, as each successful division with zero remainder indicates the presence of a rational zero. For this polynomial, the rational zeros identified are \(-2, 7,\) and \(-4\). Rational zeros, hence, are essential in breaking down polynomials into a manageable set of algebraic expressions.