Problem 20 \(f(x)=11-18 x^2-5 x^5-12 x^4-2 ... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 20

$f(x)=11-18 x^2-5 x^5-12 x^4-2 x$

Short Answer

Expert verified

The simplified form of the function is $f(x) = -5x^5 -12x^4 -18x^2 - 2x + 11$

Step by step solution

Identify Like Terms

Like terms are terms that contain the same variables raised to the same power. For the given function $f(x)=11-18x^2-5x^5-12x^4-2x$, there are no like terms. However, the powers of x are not in descending order.

Rearrange in Descending Powers

Arrange the terms so that the powers of x are in descending order. That is: $f(x) = -5x^5 -12x^4 -18x^2 - 2x + 11$

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Key Concepts

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

Like Terms

When working with polynomials, understanding like terms is crucial. Like terms are terms that have the exact same variables raised to the same powers. This means that only the coefficients (the numerical parts) differ.
For example:

The terms $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ raised to the second power.
However, $4x^3$ and $2x^2$ are not like terms because the exponents of $x$ are different.

When simplifying polynomial expressions, like terms can be combined by adding or subtracting their coefficients. In the exercise, the polynomial $f(x) = 11 - 18x^2 - 5x^5 - 12x^4 - 2x$ contains no like terms. This means each term is distinct; each term involves the variable $x$ raised to different powers.

Descending Order

Organizing the terms of a polynomial in descending order is a standard practice. This means arranging the terms starting from the highest power of the variable down to the lowest. This approach helps in better visualization and can make further operations easier to manage.
To arrange a polynomial in descending order, follow these steps:

Identify the degree of each term, which is the exponent of the variable.
Order the terms starting with the highest degree and ending with the constant term (if present).

For the given function $f(x)=11-18x^2-5x^5-12x^4-2x$, reordering gives us the polynomial $-5x^5 -12x^4 -18x^2 - 2x + 11$. This new arrangement makes it clear at a glance which term has the greatest influence at high values of $x$, and sets the stage for any derivative or integration operations that might follow.

Polynomial Function

A polynomial function is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
Crucial points regarding polynomial functions:

The highest power of the variable is known as the degree of the polynomial. It guides the behavior and properties of the polynomial function.
A polynomial's degree indicates the maximum number of potential roots or zeros it might have.
Polynomials are typically classified according to their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
Coefficients in a polynomial can be any real number and, together with the structure of the polynomial, determine its overall graph shape and critical points.

In our example, $f(x) = -5x^5 - 12x^4 - 18x^2 - 2x + 11$ is a fifth-degree polynomial because the highest power of $x$ is five. Understanding polynomial functions underpins much of algebra and calculus, as they are the building blocks for more complex expressions and equations.

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91影视

\(f(x)=11-18 x^2-5 x^5-12 x^4-2 x\)

Short Answer

Step by step solution

Identify Like Terms

Rearrange in Descending Powers

Key Concepts

Like Terms

Descending Order

Polynomial Function

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