Problem 70 - A Graphical Approach to Precalculus with Limits

Question

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=-x^{3}+4 x^{2}+7 x-28 ; 4$$

Step-by-Step Solution

Answer

The zeros are 4, \(\sqrt{7}\), and \(-\sqrt{7}\).

Step 1: Identify the given zero

We are given that one of the zeros of the polynomial \(P(x) = -x^3 + 4x^2 + 7x - 28\) is \(x = 4\). This means that \(P(4) = 0\).

Step 2: Perform synthetic division

Since \(x = 4\) is a root, perform synthetic division on \(-x^3 + 4x^2 + 7x - 28\) using \(x = 4\) to help factor the polynomial. List the coefficients: \([-1, 4, 7, -28]\).

Step 3: Divide using synthetic division

- Bring down the leading coefficient: \(-1\).- Multiply it by \(4\) (the zero), write the result below the next coefficient: \(-1 \times 4 = -4\).- Add this to the next coefficient: \(4 + (-4) = 0\).- Repeat the procedure for the next two coefficients: - Multiply \(0 \times 4 = 0\), add to \(7\): gives \(7\). - Multiply \(7 \times 4 = 28\), add to \(-28\): gives \(0\).- The result is \([-1, 0, 7, 0]\), confirming \(4\) is a root and the remainder is \(0\).- The quotient, \(-x^2 + 7\), is a quadratic expression.

Step 4: Solve the quadratic equation

The quadratic expression from our division is \(-x^2 + 7 = 0\). Move terms to solve for \(x\):\[-x^2 + 7 = 0\]\[-x^2 = -7\]\[x^2 = 7\]\[x = \pm \sqrt{7}\]

Step 5: List all zeros

The polynomial has three zeros. One given zero is \(x = 4\), and the other zeros found from the quadratic equation are \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).

Key Concepts

Synthetic Division

Synthetic Division is a simplified form of polynomial division, primarily used to divide a polynomial by a linear binomial of the form \(x - c\), where \(c\) is the given zero. It is a quicker and easier method compared to the traditional long division.

**How Synthetic Division Works:**
Synthetic Division involves using only the coefficients of the polynomial being divided. For example, if you are dividing \( -x^3 + 4x^2 + 7x - 28\) by \(x - 4\), you will only work with the coefficients \([-1, 4, 7, -28]\). Here is a brief overview of the process:

• Write the zero (\

Quadratic Equation

A Quadratic Equation is a second-degree polynomial equation in the form of \(ax^2 + bx + c = 0\). Solving quadratic equations is crucial since they often appear in various mathematical problems.

**Solving the Quadratic Equation:**
After performing Synthetic Division on the polynomial \(-x^3 + 4x^2 + 7x - 28\) using the zero \(x = 4\), we were left with quadratic \(-x^2 + 7 = 0\). To solve this equation, follow these steps:

• Move all terms to one side to set the equation to zero: \(-x^2 + 7 = 0\).
• Simplify the equation: \(-x^2 = -7\).
• Divide by \(-1\): \(x^2 = 7\).
• Take the square root of both sides to solve for \(x\): \(x = \pm \sqrt{7}\).

These solutions \(x = \sqrt{7} \) and \(x = -\sqrt{7} \) represent the remaining zeros of the polynomial.
Quadratic equations like this one can be solved using the quadratic formula, factorization, or completing the square methods when applicable. However, in our case, since the equation was simple, taking square roots was sufficient.

Roots of Polynomials

Roots of Polynomials, also known as zeros, are the values of \(x\) for which the polynomial equals zero. They are essentially the solutions to the polynomial equation.

**Understanding Polynomial Roots:**
Finding the roots of a polynomial can provide valuable insights into the graph and nature of the polynomial function. For any given polynomial \(P(x)\), the roots are the values of \(x\) such that \(P(x) = 0\).

• The polynomial \(P(x) = -x^3 + 4x^2 + 7x - 28\) has three roots because it is a cubic polynomial.
• One root, \(x = 4\), was provided in the problem.
• We found the other two roots by factoring the polynomial: \(x = \sqrt{7} \) and \(x = -\sqrt{7} \).

Knowing the number and type of roots can help us sketch the graph of the polynomial and understand its behavior. In this case, knowing three roots tells us that the graph will have three x-intersections.

Polynomial Factorization

Polynomial Factorization involves expressing a polynomial as a product of its factors, which can be simpler polynomial expressions. Factoring is an important process because it aids in finding polynomial roots and simplifies complex expressions.

**Steps in Polynomial Factorization:**
Factoring helps break down a polynomial into its simpler components, often making it easier to solve for its zeros.

• We factored the cubic polynomial \(-x^3 + 4x^2 + 7x - 28\) after discovering one root \(x = 4\).
• Synthetic Division helped us reduce the cubic polynomial to the quadratic \(-x^2 + 7\).
• We then found the roots of this quadratic to be \(x = \sqrt{7} \) and \(x = -\sqrt{7} \).
• The complete factorization of the polynomial in terms of its zeros is \((x - 4)(x - \sqrt{7})(x + \sqrt{7})\).

Factorization is a powerful technique in algebra and calculus, helping to simplify and solve various mathematical problems efficiently.