Problem 6 - Precalculus : Building Concepts and Connections

Question

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$p(x)=(x+6)^{10}, x=6,-6,0$$

Step-by-Step Solution

Answer

-6 is the only zero of the polynomial from the given list.

Step 1: Identify the Zeros of the Polynomial

Let us start by examining the polynomial \(p(x)=(x+6)^{10}\). We want to know what values of \(x\) make the polynomial equal to zero. Note that if any factor in the multiplication of a polynomial equals zero, then the polynomial equals zero. Here, we have one factor, \(x+6\), and this can equal zero when \(x = -6\). Thus, -6 is a zero of the polynomial.

Step 2: Checking the potential zeros

Now, let's substitute each of the listed numbers into the function and see if the result is zero: a) For \(x=6\), the polynomial gives \((6+6)^{10}\) which does not equal zero. b) For \(x=-6\), the polynomial gives \((-6+6)^{10}\) which equals zero. c) For \(x=0\), the polynomial gives \((0+6)^{10}\) which does not equal zero.

Step 3: Conclude the zeros that are in the given list

Based on the analysis and calculations, it's found that only \(x=-6\) leads to the polynomial equaling zero whilst being included in the given list.

Key Concepts

Polynomial Factorization

Polynomial factorization is a process used to break down a polynomial into its components, which are called factors. These components, when multiplied together, will give the original polynomial. Understanding the factorization process is crucial when trying to find the zeros of a polynomial.

For example, consider the polynomial given in the exercise, which is in factored form: \(p(x)=(x+6)^{10}\). Factorization reveals that the polynomial has just one repeated factor: \(x+6\), raised to the 10th power. This is an example of a polynomial with a repeated zero, as the zero corresponding to \(x+6\) is repeated 10 times. Recognizing that a zero will occur at the value of \(x\) that makes \(x+6=0\) simplifies the process of identifying zeros, and only requires solving a simple equation rather than dealing with the entire polynomial expression.

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of a variable. Each term is a product of a constant (called a coefficient) and a variable raised to a non-negative integer power.

In our exercise, the polynomial function is \(p(x)\). It’s a function because it describes a relationship between the input \(x\) and the output \(p(x)\). The given polynomial is a monomial because it has only one term and it’s a special case as its power is relatively high (10), and it has one variable \(x\). Most polynomial functions have more than one term and can have varying powers and coefficients, each of which can give important information about the function's graph, such as its shape, symmetry, and the number of times it intersects the x-axis, which directly relates to the number of its zeros.

Solving Polynomials

Solving polynomials involves finding values for the variable that make the polynomial equal to zero, which are known as zeros or roots. The zeros of a polynomial function correspond to the x-intercepts of the function's graph. Each zero is a solution to the equation formed by setting the polynomial equal to zero.

To solve polynomials like the one in the exercise, \(p(x)=(x+6)^{10}\), you substitute potential zeros and solve. If the substitution results in the polynomial equalling zero, then the number substituted is a zero of the polynomial. In the exercise, substituting \(x=-6\) simplifies the expression to zero, confirming that -6 is a zero. Substituting \(x=6\) or \(x=0\), does not give a zero, as it results in a positive number. The fundamental theorem of algebra asserts that a polynomial of degree \(n\) has \(n\) zeros, counting repetitions. In the case of the exercise, the polynomial is of degree 10, which means it theoretically has 10 zeros, all of which are -6 in this case due to the repeated factor.