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Chapter 2: Problem 31

Use synthetic division to divide. \(\left(4 x^{3}-9 x+8 x^{2}-18\right) \div(x+2)\)

Short Answer

Expert verified
The result of the synthetic division \((4x^3 + 8x^2 - 9x - 18) \div (x+2)\) is \(4x^2 - 9\) with a remainder of -36.

Step by step solution

01

Order the Polynomial Correctly

The first step is to order every term of the polynomial in descending power. Also, if any terms are missing, we need to write them as 0. So, in this case, we rewrite the polynomial as \(4x^3 + 8x^2 - 9x - 18\). Here, no terms are missing.
02

Set Up Synthetic Division

The next step is to set up the synthetic division. Place the numbers representing the coefficients and constants of the polynomial on a row, and then put the value that makes the divisor equal to zero on the left. For the divisor \(x + 2\), this value is \(-2\) because \(-2 + 2 = 0\).
03

Perform Synthetic Division

First, carry down the leading coefficient which is 4. Then, multiply it by the value on the left (-2) and write the result under the next coefficient. Add this number with the coefficient above it and carry this answer down. Repeat this steps until you've gone through all terms.
04

Write the Result Polynomial

The numbers on the bottom row represent the coefficients for the result polynomial. The power of the polynomial is one lower than the original, so it starts at \(x^2\) if the original started at \(x^3\). In this case, the result polynomial is \(4x^2 + 0x - 9\), which simplifies to \(4x^2 - 9\). There is a remainder of -36.

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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 similar to long division but specifically tailored for dividing polynomials. It allows us to divide a higher-degree polynomial by a lower-degree polynomial, splitting it into a quotient and a remainder. One useful method for polynomial division is synthetic division, which is a simplified and efficient way of performing polynomial division, especially when the divisor is a linear polynomial of the form \(x + c\). Synthetic division streamlines calculations by focusing on the coefficients rather than the entire polynomial, making the process faster and less prone to error.
Coefficients
In algebra, coefficients are the numerical factors that accompany the variables in a polynomial expression. For example, in the polynomial \(4x^3 + 8x^2 - 9x - 18\), the coefficients are 4, 8, -9, and -18. During synthetic division, these coefficients are organized in a row to facilitate the division process. Each coefficient represents a component of the polynomial equation. By arranging them systematically, we can perform operations directly on these numbers. This method helps in simplifying complex equations into more manageable steps, highlighting the significance of each coefficient in deriving the final result.
Remainder Theorem
The remainder theorem is a mathematical principle that states if a polynomial \(f(x)\) is divided by a linear divisor \(x - a\), the remainder of the division is \(f(a)\). This theorem is very helpful in synthetic division because it provides a quick way to find the remainder of a polynomial division. By evaluating the polynomial at a specific value, which is the root of the divisor, we can immediately determine the remainder without completing the entire division process. In practice, this means we can use the value of the divisor that makes it zero to calculate the remainder, as seen in the exercise where \(x + 2\) transformed to \(x = -2\).
Polynomial Equations
Polynomial equations consist of terms that include variables raised to whole number powers, typically with more than one term. They are used to model a variety of mathematical and real-world situations. Solving polynomial equations usually involves techniques such as factoring, graphing, or utilizing special formulas. When dividing polynomials, the aim is to express a polynomial equation as a quotient plus a remainder. By understanding polynomial equations thoroughly, students can better appreciate the purpose and application of polynomial division and the ways in which synthetic division simplifies solving certain polynomial equations.

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