Problem 69 Solve. Find a 15 th-degree pol... [FREE SOLUTION]

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Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 69

Solve. Find a 15 th-degree polynomial for which $x-1$ is a factor. Answers may vary.

Short Answer

Expert verified

A 15th-degree polynomial for which $x-1$ is a factor is $ P(x) = (x-1)x^{14} $.

Step by step solution

Understand the Requirement

We need to construct a 15th-degree polynomial. It is given that $x-1$ is a factor. This means that when $x-1$ is substituted into the polynomial, the result should be zero.

Construct the Polynomial

Since $x-1$ is a factor, we can start with $ (x-1) $ as part of the polynomial. To make it a 15th-degree polynomial, we need 14 more factors. For generality, we can choose any other factors that will make up the remaining degrees. One choice is using powers of $x$.

Define the General Form

One possible form to achieve a 15th-degree polynomial is by writing \[ P(x) = (x-1) \times x^{14} \]. $x^{14} $ ensures the highest degree term is $x^{15} $.

Verification

Verify that $x-1$ is indeed a factor by substituting $x=1 $ into $P(x).$ \[P(1) = (1-1) \times 1^{14} = 0\] This confirms the polynomial is correct.

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Key Concepts

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

Polynomial Factorization

Polynomial factorization is a method of breaking down a polynomial into simpler polynomials, which when multiplied together, give you the original polynomial. In our example, we started with the requirement that $x-1$ is a factor of a 15th-degree polynomial. This means that the polynomial can be expressed as a product that includes $x-1$. By considering the polynomial $P(x) = (x-1) \times x^{14}$, we performed polynomial factorization to achieve a polynomial of the 15th degree with $x-1$ as one of its factors.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable (typically x) in the polynomial. In simpler terms, it tells you the polynomial's 'highest' term. For example, in the polynomial $P(x) = (x-1) \times x^{14}$, the highest power of x is 15 (as $x^{14}$ multiplies with x in $x-1$). Therefore, this is a 15th-degree polynomial. Understanding the degree of a polynomial is crucial because it helps in the polynomial's factorization and also gives insight into the polynomial's behavior.

Factor Theorem

The factor theorem is a fundamental concept in algebra that links factors and roots of a polynomial. It states that $x-a$ is a factor of a polynomial if and only if $a$ is a root of the polynomial, meaning substituting $a$ into the polynomial yields zero. In our case, $x-1$ being a factor means that substituting $x=1$ into the polynomial should give zero, which we verified by substituting and confirming $P(1) = (1-1) \times 1^{14} = 0$.

Roots of Polynomial

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In our exercise, we were given that one of the roots is 1 (from the factor $x-1$). By finding other factors to multiply with $x-1$, we can ensure the polynomial is a 15th degree. In simpler terms, each root contributes to the polynomial's factors. For example, a 15th-degree polynomial can have up to 15 roots, real or complex. Identifying roots can help in polynomial factorization and in understanding the polynomial's overall structure.

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91影视

Solve. Find a 15 th-degree polynomial for which \(x-1\) is a factor. Answers may vary.

Short Answer

Step by step solution

Understand the Requirement

Construct the Polynomial

Define the General Form

Verification

Key Concepts

Polynomial Factorization

Degree of Polynomial

Factor Theorem

Roots of Polynomial

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