Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) verify the given factor(s) of the function \(f,\) (b) find the remaining factors of \(f,(\mathrm{c})\) use your results to write the complete factorization of \(f,\) and (d) list all real zeros of \(f .\) Confirm your results by using a graphing utility to graph the function. Factor(s) \((x+2)\) \((x+3)\) \((x-5),(x+4)\) \((x+2),(x-4)\) \((2 x+1)\) \((2 x-1)\) Function $$\begin{aligned} f(x)=x^{4}-4 x^{3}-15 x^{2} \\ &+58 x-40 \end{aligned}$$

Step-by-Step Solution

Answer

The valid factor of the function \(f\) is \(x-4\). The remaining factors are \(x-2\), \(x-3\), and \(x+2\). Therefore, the complete factorization of \(f\) is \(f(x) = (x-4)(x-2)(x-3)(x+2)\). The real zeros of the function \(f\) are \(x=4\), \(x=2\), \(x=3\), \(x=-2\).

Step 1: Verification of Factors

Verify if each set of factors divide the polynomial \(f(x)=x^{4}-4 x^{3}-15 x^{2}+58 x-40\) evenly by using synthetic division. Only the factor \(x-4\) yields no remainder, which means that \(x-4\) is a factor of \(f(x)\).

Step 2: Finding Remaining Factor

After dividing \(f(x)\) by \(x-4\), the result will be another polynomial \(g(x) = x^{3} - 0x^{2} + x +10 \). One must find a root of this polynomial, which is also a factor of the original polynomial. Solve the result to find the remaining factors. The remaining factors are \(x-2\), \(x-3\), \(x+2\).

Step 3: Write the Complete Factorization

The complete factorization of \(f\) is the product of all its factors. Therefore, \(f(x) = (x-4)(x-2)(x-3)(x+2)\).

Step 4: List All Real Zeros

The real zeros of a polynomial are simply the x-values where the function equals zero. For the function \(f(x) = (x-4)(x-2)(x-3)(x+2)\), set \(f(x) = 0\), and solve for \(x\). The solutions are \(x=4\), \(x=2\), \(x=3\), \(x=-2\), all of which are real numbers.

Key Concepts

Synthetic Division

Synthetic division is a streamlined process used to divide polynomials, especially handy when working with polynomials and their roots. It's a simplified form of polynomial long division and involves fewer steps. This technique is particularly useful when verifying factors of a polynomial.To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Use the root of the factor, in this case, opposite the sign of term in \(x-c\), as a divisor.
• Bring the first coefficient down as it is, and then multiply it by the divisor, placing the result under the next coefficient.
• Add the result to the coefficient above and repeat the process across all coefficients.

If the final remainder is zero, your factor divides the polynomial perfectly; hence, it is indeed a factor. In the given exercise, when we apply synthetic division with the factor \(x-4\), there is no remainder, confirming that \(x-4\) is a legitimate factor of the polynomial.

Factorization

Factorization is the process of breaking down a complex polynomial into simpler components, or factors, that when multiplied together give the original polynomial. Identifying factors helps simplify expressions and aids in solving equations. For the polynomial function \(f(x) = x^4 - 4x^3 - 15x^2 + 58x - 40\), the task is to find such linear factors.After confirming \(x-4\) as a factor, we performed synthetic division to reduce the polynomial, resulting in a cubic polynomial \(g(x) = x^3 + x + 10\). Further testing for roots helps identify additional factors.

• If you find a root, say \(x=c\), \(x-c\) is a factor.
• Using known methods like trial factors or graphing can help suggest potential roots.
• Continued factorization of the remaining polynomial will lead to the complete set of factors.

In the exercise, we identified \(x-2\), \(x-3\), and \(x+2\) as additional factors, resulting in the full factorization: \(f(x) = (x-4)(x-2)(x-3)(x+2)\).

Real Zeros

Real zeros of a polynomial are the x-values at which the polynomial evaluates to zero. They represent the solutions to the equation obtained by setting the polynomial equal to zero.To find real zeros from a factored form of a polynomial, follow these steps:

• Set the polynomial function equal to zero.
• Solve each factor in the form \(x-c=0\).
• The solutions \(x=c\) are the real zeros.

For example, from the polynomial \(f(x) = (x-4)(x-2)(x-3)(x+2)\), setting each factor to zero gives us the zeros: \(x = 4, 2, 3, ext{ and } -2\). All these are the x-values where the graph of the polynomial will intersect the x-axis, confirming that they are indeed real zeros.

Graphing Polynomial Functions

Graphing polynomial functions helps visualize the behavior of polynomials over real number intervals. Through graphing, one can easily locate zeros and analyze the behavior at and between those zeros.Here’s how to graph a polynomial function:

• Identify zeros using factors.
• Plot these zeros on the x-axis.
• Analyze end behavior by considering the leading term’s degree and coefficient.
• Plot points in between zeros to understand the curve of the graph.

For the polynomial \(f(x) = (x-4)(x-2)(x-3)(x+2)\), its zeros are \(x = 4, 2, 3, ext{ and } -2\). It will cross the x-axis at these points. Since the leading term, \(x^4\), is positive, the graph will start from above the x-axis for large negative x-values and finish above the x-axis for large positive x-values, making it easier to predict the overall shape.