Problem 12 - College Algebra

Question

a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the zero from part (b) to find all the zeros of the polynomial function. $$ f(x)=2 x^{3}-5 x^{2}+x+2 $$

Step-by-Step Solution

Answer

The zeros of the polynomial \(f(x) =2 x^{3}-5 x^{2}+x+2 \) are \(-2\), \(0.5\), and \(-1\).

Step 1: Identifying Potential Rational Zeros

First, list all possible rational zeros of the function using the Rational Root Theorem. The theorem states that any rational root, written in lowest terms, has a numerator that is a factor of the constant term (2 in this case), and a denominator that is a factor of the leading coefficient (also 2 in this case). The factors of 2 are 1, 2, -1, and -2. Using all possible combinations of these factors, the possible rational zeros for this polynomial are +/-1, and +/-2.

Step 2: Testing the Potential Zeros with Synthetic Division

Next, use synthetic division to test these potential zeroes. Assume \(P(x)\) be the polynomial to be divided and \(α\) be the root used for synthetic division. If there exists some root such that after division, the remainder is 0, then \(α\) is a root of the polynomial. When dividing the polynomial, it turns out \(-2\) is a root. This can be checked by writing \(-2\) on the left, then writing the coefficients of the polynomial on the top right (2, -5, 1, 2). Carry out the synthetic division (multiply, add, repeat). At the end, if you get a remainder of 0, that's proof that the number is indeed a root.

Step 3: Finding All Zeros Using the Factor Theorem

The result of the synthetic division is used to lower degree of the polynomial by factoring out the polynomial by \((x+α)\). This will provide a quadratic equation which can be solved to obtain the remaining zeros. After synthetic division, a quadratic equation \(2x^{2} - x - 1 = 0\) will be left. Solve this equation using the quadratic formula \(x = [-b ± sqrt(b^2 -4ac)] / 2a\). After applying the formula, the other zeros are \(0.5\) and \(-1\).

Key Concepts

Rational Root Theorem

When facing a polynomial function such as \(f(x)=2x^3-5x^2+x+2\), the Rational Root Theorem is an essential tool for identifying possible roots. According to this theorem, any rational root of a polynomial with integer coefficients is of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

In our example, the constant term is 2 and the leading coefficient is also 2, so the factor set for both is the same: \(\{1, -1, 2, -2\}\). The possible rational zeros are therefore \(\pm1\) and \(\pm2\). The Rational Root Theorem doesn't tell us which, if any, are actual zeros — it only gives us a list of candidates to test, significantly narrowing down our options.

Synthetic Division

Once we have a list of potential rational zeros, synthetic division is the next step to determine actual zeros. This process simplifies the calculation by providing a quick method for dividing a polynomial by a binomial of the form \((x-\alpha)\). It's much like long division but streamlined.

Here's how you use it for our exercise: Write down the potential zero, then the coefficients of the polynomial (2, -5, 1, 2). By following the steps of synthetic division (multiply, add, and repeat) for each candidate zero, we identify which give a remainder of zero. In this case, \(-2\) is confirmed as a zero of the polynomial \(f(x)\) because the remainder after synthetic division is zero. Essentially, if synthetic division ends with a remainder of zero, the candidate zero is indeed a root.

Factor Theorem

The Factor Theorem works hand in hand with synthetic division. It states that if a polynomial \(f(x)\) has a root at \(x=\alpha\), then it can be factored by \((x-\alpha)\). This theorem allows us to reduce the polynomial to a lower degree polynomial once we've identified a root.

After using synthetic division and finding that \(-2\) is a root, we can factor the polynomial by \((x+2)\). This leaves us with a quadratic expression, which is easier to solve. The result of the synthetic division gives us a new, simpler equation where the roots are easier to find, leading us to uncover all zeros of the original polynomial.

Quadratic Formula

Once we've got a quadratic equation after applying the Factor Theorem, we reach the final stage: solving for the remaining zeros using the quadratic formula. The formula is \(x = [-b \pm \sqrt{b^2 -4ac}] / (2a)\), which finds the solutions for any quadratic equation of the form \(ax^2+bx+c=0\).

In the simplified version of our original polynomial, we end up with a quadratic \(2x^2-x-1=0\) after factoring out \(x+2\). Applying the quadratic formula, we calculate two more zeros: \(0.5\) and \(-1\). These solutions, in combination with the zero identified through synthetic division, give us all the zeros of the polynomial function.