Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=2 x^{3}-3 x^{2}+8 x-12$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x) = 2x^3 - 3x^2 + 8x - 12\) are \(1, 2\), and \(-3\). The polynomial as a product of linear factors is \(f(x) = 2(x - 1)(x - 2)(x + 3)\). Using a graphing utility, we can verify that these are indeed the only \(x\)-intercepts of the function graph.

Step 1: Find Possible Zeros of the Function

Use the Rational Root Theorem. This theorem states that any rational root, expressed in the form \(p/q\), where p is a factor of the constant and q is a factor of the leading coefficient. For coefficients \(2\) and \(-12\), possible rational zeros would be \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).

Step 2: Verify The Roots

Substitute the possible zeros into the function until finding a value that makes the function equals zero. Checking, we find that \(1, 2\), and \(-3\) are indeed the roots of the function.

Step 3: Write the polynomial as product of Linear Factors

Having found the zeros \(1, 2\), and \(-3\), we can write the polynomial as a product of linear factors as follows: \(f(x) = 2(x - 1)(x - 2)(x + 3)\). According to Factor Theorem, a polynomial \(f(x)\) has a factor \((x - k)\) if and only if \(f(k) = 0\).

Step 4: Determine the \(x\)-intercepts of the Function Graph

The \(x\)-intercepts of the graph of the function are the real roots of the function, which we already determined as \(1, 2\), and \(-3\).

Step 5: Verifying the Zeros using Graphing Utility

Use a graphing utility to graph the function and verify that the real zeros determined earlier are the only \(x\)-intercepts. The graph should intersect the \(x\)-axis at \(1, 2\), and \(-3\) if the roots are correct.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool used for finding possible rational zeros of a polynomial. It simplifies our search for zeros by providing a finite list of possible candidates. For a polynomial like \(f(x) = 2x^3 - 3x^2 + 8x - 12\) with integer coefficients, the theorem states that any potential rational zero can be expressed as \(\frac{p}{q}\):

• \(p\) is a factor of the constant term, which is \(-12\)
• \(q\) is a factor of the leading coefficient, which is \(2\)

This means, for our example, the potential rational zeros are \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 4\), \(\pm 6\), and \(\pm 12\). Once we have this list, we can test each to see if substituting them into the polynomial results in zero, thus confirming them as roots.

Factor Theorem

The Factor Theorem is closely related to the concept of polynomial division. It states that a polynomial \(f(x)\) has a factor \((x - k)\) if and only if \(f(k) = 0\). This helps in breaking down the polynomial into linear factors if we already know the polynomial's zeros or roots.
When we revisited the polynomial \(f(x) = 2x^3 - 3x^2 + 8x - 12\) and found its zeros as \(1\), \(2\), and \(-3\), each zero corresponds to a factor. Thus, we can express the polynomial as the product \(f(x) = 2(x - 1)(x - 2)(x + 3)\), indicating that \((x - 1)\), \((x - 2)\), and \((x + 3)\) are the linear factors.

Graphing Polynomial Functions

Graphing polynomial functions gives a visual insight into the behavior of the function. By identifying the zeros, we understand where the graph intersects the \(x\)-axis. For our polynomial \(f(x) = 2x^3 - 3x^2 + 8x - 12\), with known zeros \(1\), \(2\), and \(-3\), the graph confirms these as the \(x\)-intercepts.
Graphing utilities like graphing calculators or software help verify these zeros. The tool displays the polynomial curve, allowing us to visually confirm that the graph touches or crosses the \(x\)-axis at the calculated zeros. This provides a check for our earlier algebraic work, ensuring all computed zeros are indeed accurate and real.

Linear Factors

Linear factors reflect simpler expressions that, when multiplied, recreate the original polynomial. Each linear factor corresponds to a root of the polynomial. For the polynomial \(f(x) = 2x^3 - 3x^2 + 8x - 12\), whose zeros are \(1\), \(2\), and \(-3\), we can break it into linear factors. This is done as follows:

• \((x - 1)\) because \(f(1)=0\)
• \((x - 2)\) because \(f(2)=0\)
• \((x + 3)\) because \(f(-3)=0\)

Therefore, the polynomial can be expressed as \(f(x) = 2(x - 1)(x - 2)(x + 3)\). This factorization is essential for understanding the structure of the polynomial and solving related problems.