Problem 23 Use synthetic division to divide... [FREE SOLUTION]

Chapter 2: Problem 23

Use synthetic division to divide the first polynomial by the second. $$x^{6}+x-10, \quad x+3$$

Short Answer

Expert verified

The resulting polynomial after dividing $x^{6}+x-10$ by $x+3$ using synthetic division is $x^{5} -3x^{4} +6x^{3} -18x^{2} +54x -81$ with a remainder of 246.

Step by step solution

Setup

First, write down the coefficients of the dividend polynomial in decreasing powers, from left to right. If there are missing powers in the polynomial, enter zero for those coefficients. The divisor is $x+3$, so the number it can equal to zero is -3. The synthetic division table then looks like this:\n -3 | 1 0 0 0 0 1 -10

Execute the synthetic division

Look at the first number of the coefficients, which is 1. Drag this number straight down. Multiply -3 (from the divisor) by the number just dragged down (1). Write this result (-3) underneath the next number (0) of the coefficients. Add these two numbers together(0-3=-3) and write the result directly beneath them (-3). Continue this process for each of the coefficients: \n -3 | 1 0 -3 9 -27 81 -246 \n-----------------------\n 1 -3 6 -18 54 -81 246

Write the result

The final result can be written by starting with x to one less power than the original dividend (since we divided by x+2, which is first degree) and the last number as the remainder. So the final answer is: $x^{5} -3x^{4} +6x^{3} -18x^{2} +54x -81, remainder 246$

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Key Concepts

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

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to long division with numbers, but here we deal with variables. In our example, we divided a sixth-degree polynomial, $x^6 + x - 10$, by a simpler linear polynomial, $x + 3$. This division aims to simplify complex polynomials by breaking them down, much like dividing numbers into smaller parts.

It is often used to simplify expressions.
There are different methods, like long and synthetic division.
Choosing the right method depends on the specific problem.

Typically, polynomial division provides a quotient and sometimes a remainder, just like dividing an integer does.

Remainder Theorem

The Remainder Theorem provides a quick way to find the remainder of a polynomial division without performing the complete division process. According to the theorem, if you divide a polynomial $f(x)$ by a linear polynomial $x-c$, the remainder of this division is simply $f(c)$.
In the solution, we used synthetic division, and our remainder was 246. This theorem is very useful to quickly evaluate polynomials at given points and check divisibility:

If $f(c) = 0$, then $x-c$ is a factor of $f(x)$.
The theorem saves time in calculations and insights into polynomial properties.

Applying this theorem can greatly enhance understanding of polynomial behavior and divisibility quickly with minimal computation.

Polynomials

Polynomials are expressions composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They represent a wide range of functions and equations in mathematics and science.
In our given problem, the expression $x^6 + x - 10$ is a polynomial. Key characteristics of polynomials include:

Their terms are ordered by decreasing powers.
The coefficients can be whole numbers, decimals, or fractions.
They can have one or more variables.

Polynomials are foundational in algebra and calculus, and understanding their properties is crucial for solving complex mathematical problems.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It helps in understanding the polynomial's behavior and its complexity.
For instance, in our polynomial example $x^6 + x - 10$, the highest power of $x$ is 6, which makes it a sixth-degree polynomial. Knowing the degree is essential for several reasons:

It indicates the maximum number of roots a polynomial can have.
Determines the polynomial's end behavior.
Helps in selecting appropriate methods for solving or factoring it.

Understanding the degree also aids in graphing polynomials and predicting their shapes and intersections with axes. This makes it a vital concept in algebra and higher mathematics.

91影视

Use synthetic division to divide the first polynomial by the second. $$x^{6}+x-10, \quad x+3$$

Short Answer

Step by step solution

Setup

Execute the synthetic division

Write the result

Key Concepts

Polynomial Division

Remainder Theorem

Polynomials

Degree of a Polynomial

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