Problem 5 - Precalculus : Building Concepts and Connections

Question

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$p(x)=(x-10)^{8}, x=6,-10,10$$

Step-by-Step Solution

Answer

The given polynomial \(p(x)=(x-10)^8\) has only one root, x=10. The rest of the values x=6 and x=-10 are not roots of the polynomial.

Step 1: Verification of x=6

Substitute \(x=6\) into the polynomial \(p(x)=(x-10)^8\), we get \[p(6)=(6-10)^8=4^8\neq0\]. So, x=6 is not a root of the polynomial.

Step 2: Verification of x=-10

Now substitute \(x=-10\) into the polynomial, we get \[p(-10)=(-10-10)^8=20^8\neq0\]. So, x=-10 is also not a root of the polynomial.

Step 3: Verification of x=10

Finally, substituting \(x=10\) into the polynomial, we get \[p(10)=(10-10)^8=0\]. Therefore, x=10 is a root of the polynomial.

Key Concepts

Polynomial Roots

Understanding polynomial roots is essential for solving various mathematical problems involving polynomials. These roots are also known as zeroes of the polynomial. They represent the values of the variable for which the polynomial evaluates to zero. In simpler terms, when a particular value is substituted for the variable in a polynomial, and the result is zero, that value is termed a root of the polynomial.

For instance, in the exercise where we investigate the polynomial \(p(x) = (x-10)^{8}\), when we substitute the value \(x=10\), \(p(x)\) becomes zero, which shows that \(x=10\) is indeed a root of the polynomial. This is confirmed through substitution and simplification in the step by step solution provided. To understand this better, it's crucial to grasp the concept of substituting the given values into the polynomial and simplifying to check whether the result is zero, which is indicative of a root.

Polynomial Function

A polynomial function is a mathematical expression involving a variable raised to whole number powers and co-efficients. These functions are written in the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \), where the powers of \(x\) are called degrees, and the \(a_i\)'s - where \(i\) ranges from 0 to \(n\) - are the coefficients. The degree of the polynomial is the highest power of the variable \(x\) that appears in the function. In the given exercise, \(p(x) = (x-10)^{8}\) is a polynomial function of a single variable \(x\) and has a degree of 8.

The behavior of the polynomial function greatly depends on its degree and leading coefficient. For instance, if the degree is even and the leading coefficient is positive, the graph of the polynomial has a similar shape to a parabola opening upwards; while for odd degrees, the end behavior of the graph will mimic the letter 'S' or its reverse. Polynomial functions play an integral role in various applications, including physics, engineering, and economics.

Exponentiation

Exponentiation is an arithmetic operation involving two numbers, the base and the exponent. It is a way to express prolonged multiplication of the same factor and is written as \(b^n\), where \(b\) is the base and \(n\) is the exponent or power. The exponent denotes how many times the base is multiplied by itself. For example, \(3^4\) means multiplying 3 by itself 4 times: \(3 * 3 * 3 * 3\).

In the context of the exercise, we encounter exponentiation in the polynomial \(p(x) = (x-10)^{8}\), which implies \(x-10\) multiplied by itself 8 times. It's important to understand exponentiation because it influences the behavior of the function and its roots. For example, the exponent here suggests that, although \(x=10\) is the only real root, it is a root of multiplicity 8, indicating that the curve of the function will just touch the x-axis at the point \(x=10\) and turn back instead of crossing it. Exponentiation can greatly affect the number and type of roots a polynomial has, and it is a key concept in understanding the structure of polynomial functions.