Problem 9 - College Algebra

Question

a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the zero from part (b) to find all the zeros of the polynomial function. $$ f(x)=x^{3}+x^{2}-4 x-4 $$

Step-by-Step Solution

Answer

The zeros of the polynomial function \(f(x)=x^{3}+x^{2}-4x-4\) are -2, -1 with multiplicity 2.

Step 1: List possible rational zeros

Possible rational zeros for a polynomial with integer coefficients can be calculated as the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is -4 and the leading coefficient is 1. Therefore, the factors of -4 are ±1, ±2, ±4, and the factors of 1 is just ±1. So, the possible rational zeros of \(f(x)\) are just these factors i.e. ±1, ±2, ±4.

Step 2: Use synthetic division to test possible zeros

Using synthetic division, test the possible rational zeros from step 1 until you find a zero remainder, indicating that you've found an actual zero of the polynomial. Starting with 1, x=1 gives the remainder -6 which is not zero. Therefore, 1 is not a zero for the polynomial. Similarly, test values -1, 2 and -2. For -2, the synthetic division gives a remainder of zero indicating that x=-2 is a zero to the polynomial.

Step 3: Find all the zeros of the function

The synthetic division in step 2 gives the quotient \(x^2 + 3x + 2\), this indicates that the original function can be written as \((x+2)(x^{2}+3x+2)\). Equating \(x^{2}+3x+2\) to zero, we solve for x to find the remaining zeros. Factoring the quadratic equation, we get \((x+2)(x+1)^{2}=0\). The remaining zeros of the function are x=-1 (double root). So, all the zeros for the given polynomial are -2 and -1 (with multiplicity 2).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, particularly useful when dividing by a linear divisor. It's a quicker and more compact alternative to the long division of polynomials. Here's how it works:

• First, select a potential zero from the list of rational zeros, which are the possible roots of the polynomial.
• Arrange the coefficients of the polynomial in descending order of the terms.
• Use the selected zero and bring down the leading coefficient.
• Multiply this coefficient by the chosen zero and add the result to the next coefficient. Continue this process until all coefficients have been used.
• If the last number (the remainder) is zero, then the chosen potential zero is indeed a zero of the polynomial, and the quotient is our newly formed polynomial.

In our exercise, we applied synthetic division using -2, a potential zero, to find a zero remainder, confirming x = -2 as a zero of the polynomial function.

Polynomial Function Zeros

Zeros of a polynomial function are values for which the function equals zero. They are crucial as they give us the solutions to the equation formed when a polynomial is set to zero.

• Finding these zeros involves solving the equation after dividing the polynomial using methods like synthetic division.
• They help factor the polynomial further.
• Zeros can be real or complex numbers, with real zeros being visible as points where the graph of the function crosses or touches the x-axis.
• The multiplicity of a zero refers to how many times that zero fulfills the polynomial equation.

In this exercise, the zeros of the polynomial function \(f(x) = x^3 + x^2 - 4x - 4\) were found to be x = -2, and x = -1 with multiplicity 2. These represent the points where the polynomial equals zero.

Factorization of Polynomials

Factorizing a polynomial involves breaking it down into the product of its factors. Factors are polynomial expressions that, when multiplied, yield the original polynomial.

• Once zeros are found using methods such as synthetic division, these can guide us in rewriting the polynomial as a product of factors.
• Each zero contributes to a factor of the form \((x - \, \text{zero})\).
• In our polynomial \(f(x) = x^3 + x^2 - 4x - 4\), once the first zero was found at x = -2, this prompted us to express the polynomial as \((x+2)\) times the quotient polynomial \(x^2 + 3x + 2\).
• Further factorization of \(x^2 + 3x + 2\) eventually resulted in \((x+2)(x+1)^2\), demonstrating a complete factorization.

Factorization not only helps in solving polynomial equations but also provides insights into the geometric nature of the polynomial's graph and its intersections with the x-axis.