Problem 65 - A Graphical Approach to College Algebra

Question

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=3 x^{3}+5 x^{2}-3 x-2 ;-2$$

Step-by-Step Solution

Answer

Additional zeros are \(-\frac{1}{3}\) and \(1\).

Step 1: Verify Given Zero

We are given that \(-2\) is a zero of the polynomial \(P(x) = 3x^3 + 5x^2 - 3x - 2\). To verify, substitute \(x = -2\) into \(P(x)\). Calculate \(P(-2) = 3(-2)^3 + 5(-2)^2 - 3(-2) - 2 = 0\). Since \(P(-2) = 0\), \(-2\) is confirmed as a zero of \(P(x)\).

Step 2: Use Synthetic Division

Perform synthetic division of the polynomial \(P(x)\) by \((x + 2)\) to reduce the degree of the polynomial. Set up synthetic division using the coefficients \(3, 5, -3, -2\). The result of the division will be \(3x^2 - x - 1\), with no remainder, confirming that \(x + 2\) is a factor.

Step 3: Factor Quadratic Polynomial

Now, factor the quotient from the synthetic division, \(3x^2 - x - 1\), to find its zeros. We can factor it by splitting the middle term:\[3x^2 - 3x + 2x - 1 = (3x + 1)(x - 1)\].

Step 4: Find Remaining Zeros

The remaining zeros of the polynomial are the zeros of \(3x + 1 = 0\) and \(x - 1 = 0\). Solving these, we find \(x = -\frac{1}{3}\) and \(x = 1\), respectively.

Key Concepts

Synthetic Division

Synthetic Division is a shortcut method used for dividing a polynomial by a binomial of the form \(x - k\). It is primarily utilized to find out whether a given value is a zero of the polynomial and to simplify polynomial expressions. In our case, the polynomial is \(P(x) = 3x^3 + 5x^2 - 3x - 2\) and the given zero is \(-2\). Performing synthetic division with this zero effectively reduces the degree of the polynomial, making it easier to manage.
To begin, you need to arrange the coefficients of the polynomial: \(3, 5, -3, -2\). Setting up the synthetic division involves writing these coefficients in a row and bringing down the leading coefficient (first number) under the line. Multiply the brought down number by your zero, in this case \(-2\). Add this result to the next coefficient. Repeat these steps for all coefficients.
For our polynomial when we divide using \(x + 2\), we transition from coefficients \(3, 5, -3, -2\) to the resulting coefficients \(3, -1, -1\), representing the quotient \(3x^2 - x - 1\). The zero remainder confirms that \((x + 2)\) is indeed a factor of the polynomial.

Quadratic Factorization

After reducing the polynomial through synthetic division, we're left with a quadratic expression \(3x^2 - x - 1\). To find more zeros of the original polynomial, this quadratic needs to be factored into simpler expressions. Factoring a quadratic often requires finding two numbers whose product equals the product of the first and last coefficients and whose sum equals the middle coefficient.
For \(3x^2 - x - 1\), you split the middle term \(-x\) into \(-3x + 2x\), allowing you to group terms. This gives us \((3x^2 - 3x) + (2x - 1)\). Factoring by grouping leads to the expression \((3x + 1)(x - 1)\).
The process of quadratic factorization helps identify potential zeros, turning a complex polynomial problem into simpler linear factors that are easy to solve.

Polynomial Roots

Polynomial Roots, also known as zeros, are values that satisfy the equation \(P(x) = 0\). Finding these roots is crucial because they reveal significant properties of the polynomial function. Once a polynomial is factored, the roots can be easily obtained by setting each factor equal to zero and solving for \(x\).
From our quadratic factorization \((3x + 1)(x - 1)\), the polynomial roots can be computed by assuming each factor equal to zero. Solving \(3x + 1 = 0\) yields \(x = -\frac{1}{3}\) and solving \(x - 1 = 0\) gives \(x = 1\). Including the given zero \(-2\), the complete set of zeros for \(P(x)\) is \(-2, -\frac{1}{3}, 1\).
Understanding polynomial roots is essential as they provide insight into the behavior of the graph of the polynomial, including where it intercepts the x-axis.