Problem 70 - A Graphical Approach to College Algebra

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: Use the Given Zero to Form a Factor

We are given that one of the zeros of the polynomial is 4. This means that \(x - 4\) is a factor of the polynomial \(P(x)\). Therefore, we can express the polynomial as \(P(x) = (x - 4)Q(x)\), where \(Q(x)\) is the quotient polynomial.

Step 2: Perform Polynomial Division

To find \(Q(x)\), we need to divide \(P(x) = -x^3 + 4x^2 + 7x - 28\) by \(x - 4\) using synthetic or long division. 1. The division yields: \(-x^3 + 4x^2 + 7x - 28 \div (x - 4) = -x^2 + 7\).Thus, \(P(x) = (x - 4)(-x^2 + 7)\).

Step 3: Solve the Quadratic Factor

We now have the expression \(P(x) = (x - 4)(-x^2 + 7)\). To find the remaining zeros, solve the quadratic equation \(-x^2 + 7 = 0\) for \(x\).1. Rearrange the quadratic equation: \(-x^2 = -7\) \(x^2 = 7\)2. Take the square root of both sides to solve for \(x\): \(x = \pm \sqrt{7}\).

Step 4: List All Zeros of the Polynomial

The polynomial \(P(x) = -x^3 + 4x^2 + 7x - 28\) has three zeros: 4, \(\sqrt{7}\), and \(-\sqrt{7}\). These correspond to the factors \(x - 4\), \(x - \sqrt{7}\), and \(x + \sqrt{7}\) in the factorization of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is like long division with numbers but with polynomials. This process helps us divide one polynomial by another to find a quotient and a remainder. In our problem, we use polynomial division to divide the polynomial \( P(x) = -x^3 + 4x^2 + 7x - 28 \) by \( x - 4 \). By doing this, we can express the original polynomial in the form of \( P(x) = (x - 4)Q(x) \), where \( Q(x) \) is the quotient.

• The dividend is the polynomial you start with, \(-x^3 + 4x^2 + 7x - 28\).
• The divisor is \(x - 4\), the factor that represents a zero of the polynomial.

Utilizing polynomial division, we discover \(Q(x)\), making complex polynomials easier to handle, especially when finding zeros.

Synthetic Division

Synthetic division is a simplified form of polynomial division. It's faster and more straightforward compared to long division, especially when dividing by a linear factor like \(x - 4\). It is particularly useful when you know one factor of the polynomial.

• Begin by setting up the coefficients of \(P(x)\): \(-1, 4, 7, -28\).
• Use the zero, 4, to perform the operation.

This process results in the quotient polynomial which, in our case, turns out to be \(-x^2 + 7\). Synthetic division facilitates this by eliminating the repetitive process of longer division, providing a quick path to the quotient.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically in the form \(ax^2 + bx + c = 0\). Solving these gives us the zeros. Once we have split the polynomial by division, we are left with the quadratic part: \(-x^2 + 7\). When we set this piece equal to zero, \(-x^2 + 7 = 0\), we can solve for the remaining zeros:

• Rearrange to solve for \(x^2\): \(x^2 = 7\).
• Take the square root: \(x = \pm \sqrt{7}\).

This method of solving allows us to find any remaining zeros efficiently.

Factorization

Factorization involves breaking down an expression into a product of simpler expressions. For a polynomial, finding factors helps in finding its zeros. In our polynomial \( P(x) = -x^3 + 4x^2 + 7x - 28 \), the given zero and division help in the initial factorization: \(P(x) = (x - 4)(-x^2 + 7)\).

• The linear factor \(x - 4\) stems from the given zero, \(4\).
• The quadratic factor \(-x^2 + 7\) arises from the division process.

With factorization complete, analyzing and solving the polynomial becomes more approachable, enabling us to list all zeros: \(4\), \(\sqrt{7}\), and \(-\sqrt{7}\).