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Magazine Chapter 3: Problem 390
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ f(x)=3 x^{3}+x^{2}-20 x+12 ; x+3 $$
Short Answer
Expert verified
The real zeros are \(x = -3, \frac{2}{3}, 2\).
Step by step solution
01
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.
02
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 \(3x^2 - 8x + 4\).
03
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 \((x - 2)\): \[ (3x - 2)(x - 2). \]
04
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 \(x = -3, \frac{2}{3}, 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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鈥檚 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.
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.
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.
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.
