Problem 16 $$ x^{5}-7 x^{4}+16 x^{3}-16 x... [FREE SOLUTION]

Chapter 7: Problem 16

$$ x^{5}-7 x^{4}+16 x^{3}-16 x^{2}+7 x-1 $$

Short Answer

Expert verified

The given polynomial is $x^5 - 7x^4 + 16x^3 - 16x^2 + 7x - 1$, with a degree of 5 and six terms. When evaluating for specific values of x, we found that for $x=0$ and $x=1$, the polynomial equals 0. To better understand the polynomial, students can graph the function to identify its shape and zeroes.

Step by step solution

Identify the degree and terms of the polynomial

The given polynomial is $$x^5-7x^4+16x^3-16x^2+7x-1$$. The highest power of x in this expression is 5, so the degree of the polynomial is 5. The expression has six terms: $$x^5$$, $$-7x^4$$, $$16x^3$$, $$-16x^2$$, $$7x$$, and $$-1$$.

Evaluate the polynomial for specific values of x

Now, let's substitute a few values for x in the polynomial expression and evaluate the polynomial for each value. -- Value of x = 0 $$f(0)= (0)^5 -7(0)^4 + 16(0)^3 - 16(0)^2 + 7(0)-1 = 0$$ -- Value of x = 1 $$f(1)= (1)^5 -7(1)^4 + 16(1)^3 - 16(1)^2 + 7(1)-1 = 1 - 7 + 16 - 16 + 7 - 1 = 0$$ You can continue evaluating the polynomial for other values of x.

Graph the polynomial

To get a visual representation of the polynomial, students can graph the function $ f(x) = x^5 - 7x^4 + 16x^3 - 16x^2 + 7x - 1 $ using graphing calculators or online graphing tools. The graph will help students understand the shape of the polynomial and its zeroes (where the graph crosses the x-axis).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of Polynomial

The degree of a polynomial is a key concept in understanding its behavior and is determined by the highest power of the variable in the expression. For the polynomial given in the exercise, which is $ x^5 - 7x^4 + 16x^3 - 16x^2 + 7x - 1 $, the highest power of $ x $ is 5. Thus, this polynomial is said to be of degree 5.

Knowing the degree of a polynomial provides insight into several important characteristics, such as:

The number of roots or solutions the polynomial can have. A polynomial of degree $ n $ can have up to $ n $ roots.
The basic structure of its graph, including the number of turning points it can have, which is up to $ n-1 $.
Its end behavior, where the graph will tend to rise to infinity or fall to negative infinity, depending on the leading coefficient. For instance, if the leading term's coefficient is positive, the ends of the graph rise as $ x $ approaches positive or negative infinity.

Understanding these aspects can help in tackling more complex problems involving polynomial functions.

Evaluating Polynomials

Evaluating a polynomial involves calculating its value by substituting specific values for $ x $. This can help determine particular points on the graph of the polynomial or verify if certain numbers are roots.

For the polynomial $ f(x) = x^5 - 7x^4 + 16x^3 - 16x^2 + 7x - 1 $, let's break down the steps when $ x = 1 $:

Replace $ x $ in the polynomial with 1: \[ f(1) = 1^5 - 7(1)^4 + 16(1)^3 - 16(1)^2 + 7(1) - 1 \]
Simplify the expression: $ 1 - 7 + 16 - 16 + 7 - 1 = 0 $.

So, $ f(1) = 0 $, indicating that $ x = 1 $ is a root of the polynomial.

By continuing this method with other values, one can decipher more roots, find intersection points or gain understanding of polynomial behavior at specific X-values.

Graphing Polynomials

Graphing polynomials provides a powerful visual understanding of how the function behaves across different ranges of $ x $. For a polynomial such as $ f(x) = x^5 - 7x^4 + 16x^3 - 16x^2 + 7x - 1 $, graphing can reveal vital information such as:

The degree and leading coefficient dictate the graph's end behavior. Here, since it is a fifth-degree polynomial, it should begin and end pointing in opposite directions because the leading coefficient is positive.
The points where the polynomial crosses the x-axis represent its roots. Evaluating or approximation methods may provide these points. In this polynomial, we see it has one root already identified at $ x = 1 $.
The graph's turning points, which are related to the degree, providing up to four turning points for this fifth-degree polynomial.

Using graphing calculators or online graphing tools, plotting these points and observing their connections forms the full graph, affording insight into both apparent and subtle features of the polynomial. It's like drawing a bridge between algebraic expressions and their geometrical representations, enhancing comprehension and application.

91影视

$$ x^{5}-7 x^{4}+16 x^{3}-16 x^{2}+7 x-1 $$

Short Answer

Step by step solution

Identify the degree and terms of the polynomial

Evaluate the polynomial for specific values of x

Graph the polynomial

Key Concepts

Degree of Polynomial

Evaluating Polynomials

Graphing Polynomials

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