Problem 10 $f(x)=x^4+5 x^3-7 x^2-29 x+30$... [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 10

$f(x)=x^4+5 x^3-7 x^2-29 x+30$

Short Answer

Expert verified

This is a fourth degree polynomial function. For solving, apply synthetic division and find roots of the function. The roots are the values of x where the value of this function would be zero.

Step by step solution

Determine the Degree of the Polynomial

The degree of the polynomial $f(x)=x^4+5 x^3-7 x^2-29 x+30$ can be determined by the highest exponent of the variable $x$ in the function. Here, the highest exponent is 4, so it is a fourth degree polynomial function.

Synthetic Division

Use synthetic division to simplify the polynomial function. Synthetic division is a shorthand method of performing long division in the case of a polynomial by a linear factor. It is most commonly used when finding the roots of a polynomial. Try factors of the constant term (i.e., 30 in this case) as potential roots. For a polynomial $ax^4 + ... + z$, potential roots can be 卤 factors of $z/a$. Here, potential roots can be 卤 factors of $30/1 = 30$.

Find the Roots of the Function

From synthetic division, determine roots of the equation $f(x) = x^4+5 x^3-7 x^2-29 x+30 = 0$. Roots are the values of $x$ for which the equation will be satisfied or the function crosses the x-axis on a graph. Also, for a n-degree polynomial, there will be at most n roots. Here, it will be possible to find at most four roots.

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Key Concepts

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

Synthetic Division

Synthetic division is a powerful tool that simplifies dividing a polynomial by a binomial of the form $x - c$. It condenses long division into a more manageable process and is particularly useful for finding roots of polynomials.

Here's how synthetic division works:

Select a potential root from the factors of the constant term.
List the coefficients of the polynomial. For $f(x)=x^4+5x^3-7x^2-29x+30$, the coefficients are $[1, 5, -7, -29, 30]$.
Use the potential root and these coefficients to perform synthetic division.

This method will give you either a remainder of zero, indicating that the chosen root is indeed a root of the polynomial, or a non-zero remainder, implying that it is not a root.

In real applications, finding one valid root simplifies the equation, allowing you to break down the polynomial further to find additional roots.

Polynomial Degree

The polynomial degree is an essential concept in understanding a polynomial function's behavior and graph. It is defined by the highest power of the variable $x$ present in the polynomial.

For example, in $f(x)=x^4+5x^3-7x^2-29x+30$, the degree is 4 because the term $x^4$ has the highest exponent.

Understanding the degree helps in multiple ways:

It indicates the maximum number of roots the polynomial can have.
It suggests how the graph behaves as $x$ moves towards positive or negative infinity鈥攕omething known as the end behavior.
The degree also affects the number of turning points the graph can have (one less than the degree).

Recognizing these patterns is crucial as it builds a foundation for solving polynomials and understanding higher mathematics.

Roots of Polynomials

Roots of a polynomial, also known as zeros, are the values of $x$ that make the polynomial equal to zero. In the context of a graph, they are the x-coordinates where the curve intersects the x-axis.

The process of finding roots usually begins with methods like synthetic division or factoring, particularly when seeking solutions to higher degree polynomials, such as quadratic, cubic, or quartic as in our given example.

Key points to remember about roots:

A polynomial with degree $n$ will have at most $n$ roots. In this case, the polynomial $f(x)$ can have up to four roots.
Roots can be real or complex numbers. Real roots are visible on the graph; complex roots are not.
Finding all roots sometimes requires successive factorization, converting the polynomial into simpler polynomials or using numerical methods for approximation.

Understanding roots is crucial as they provide meaningful insights into the solutions of polynomial equations and their behavior over different ranges of $x$.

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91影视

\(f(x)=x^4+5 x^3-7 x^2-29 x+30\)

Short Answer

Step by step solution

Determine the Degree of the Polynomial

Synthetic Division

Find the Roots of the Function

Key Concepts

Synthetic Division

Polynomial Degree

Roots of Polynomials

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