Problem 390 - Precalculus

Question

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ f(x)=3 x^{3}+x^{2}-20 x+12 ; x+3 $$

Step-by-Step Solution

Answer

The real zeros are \(x = -3, \frac{2}{3}, 2\).

Step 1: Apply the Factor Theorem

According to the Factor Theorem, if \(x+3\) is a factor of \(f(x)\), then \(f(-3) = 0\). Substitute \(-3\) into \(f(x) = 3x^3 + x^2 - 20x + 12\) to check this: \[ f(-3) = 3(-3)^3 + (-3)^2 - 20(-3) + 12 = -81 + 9 + 60 + 12 = 0. \]Thus, \(x+3\) is confirmed as a factor.

Step 2: Perform Synthetic Division

Now, divide the polynomial by \(x+3\) using synthetic division. The coefficients of \(f(x)\) are \(3, 1, -20, 12\).1. Write \(-3\) from \(x+3\) and set up the coefficients: \[ \begin{array}{r|rrrr} -3 & 3 & 1 & -20 & 12 \ & & -9 & 24 & -12 \ \hline & 3 & -8 & 4 & 0 \ \end{array} \]2. Bring down the first coefficient (3). Multiply by -3 and add below the second coefficient. Repeat for all subsequent coefficients.3. The remainder is 0, confirming \(x+3\) is a factor, and the quotient is \(3x^2 - 8x + 4\).

Step 3: Factor the Quotient Polynomial

Factor the quadratic \(3x^2 - 8x + 4\). To factor, we look for two numbers that multiply to \(3 \times 4 = 12\) and add to \(-8\). These numbers are \(-6\) and \(-2\).- Rewrite: \(3x^2 - 6x - 2x + 4\).- Group terms: \[ (3x^2 - 6x) + (-2x + 4) = 3x(x - 2) - 2(x - 2). \]- Factor out the common term \((x - 2)\): \[ (3x - 2)(x - 2). \]

Step 4: Find All Real Zeros

Use the factored form to find the zeros: - From \(x+3\), we have \(x = -3\).- From \((3x-2)\), set \(3x-2 = 0\): \[ 3x = 2 \Rightarrow x = \frac{2}{3}. \]- From \((x-2)\), set \(x-2 = 0\): \[ x = 2. \]The real zeros are \(x = -3, \frac{2}{3}, 2\).

Key Concepts

Synthetic Division

Synthetic Division is a fast and efficient way to divide a polynomial by a binomial of the form \(x - c\). Unlike long division, synthetic division simplifies the process by focusing only on the coefficients of the polynomial. Here’s how it works:

• Identify \(c\) from the factor \(x+c\). Here, \(c = -3\).
• List the coefficients of the polynomial. For \(3x^3 + x^2 - 20x + 12\), the coefficients are \(3, 1, -20, 12\).
• Write the number \(c\) to the left and set up the coefficients to the right in a row.
• Bring down the leading coefficient as is.

Multiply and add the result to the next coefficient, continuing to the end. At the end, if the remainder is 0, the binomial is a factor of the polynomial. If not, then it isn’t. In this problem, the remainder is 0, confirming that \(x+3\) is indeed a factor.

Polynomial Function

A polynomial function is an expression involving a sum of powers of a variable multiplied by coefficients. The general form of a polynomial function is:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \,\ldots\, + a_1x + a_0\]where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants and \(n\) represents the degree of the polynomial.
The polynomial provided in this exercise is \(f(x) = 3x^3 + x^2 - 20x + 12\). This is a cubic polynomial because its highest degree is 3.
Polynomial functions can have real and complex zeros, or roots, which are the values of \(x\) for which the function \(f(x) = 0\). These are crucial in understanding the behavior of the polynomial on a graph.

Real Zeros

Real zeros of a polynomial are the \(x\)-values where the polynomial equals zero. These zeros are the points where the graph of the polynomial crosses or touches the \(x\)-axis. To find these zeros, one can use methods like factoring, using the quadratic formula, or applying the Factor Theorem.

• Step 1: Use the Factor Theorem to check if a given binomial is a factor by substituting the value into the polynomial and checking if it yields zero.
• Step 2: Factor the polynomial using techniques like synthetic division when one factor is known.

In this exercise, using the Factor Theorem and synthetic division, the polynomial \(f(x) = 3x^3 + x^2 - 20x + 12\) was shown to have factors \((x+3)(3x-2)(x-2)\). Thus, the real zeros are \(x = -3, \frac{2}{3},\) and \(2\). These are the points at which the polynomial touches or crosses the \(x\)-axis.

Quadratic Factorization

Quadratic factorization involves breaking down a quadratic polynomial into simpler terms that are multiplied together to give back the original polynomial. A typical quadratic polynomial is in the form \(ax^2+bx+c\), which can be factored by:

• Finding two numbers \(m\) and \(n\) such that \(mn = ac\) and \(m+n = b\).
• Rewriting the polynomial by splitting \(bx\) into \(mx + nx\).
• Grouping and factoring by pairs.

In the solution, \(3x^2 - 8x + 4\) is factored as \((3x-2)(x-2)\). Finding the correct factors requires practice in spotting which numbers effectively split the middle term for clean factorization.