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Chapter 5: Problem 50

When dividing a polynomial by a binomial, explain when to stop dividing.

Short Answer

Expert verified
The division of a polynomial by a binomial ends when the degree of the remaining part of the dividend becomes less than the degree of the divisor. At this point, what remains is referred to as the remainder of the division.

Step by step solution

01

Understanding Polynomial and Binomial

In this case, the polynomial is the dividend (the quantity that you're dividing) and the binomial is the divisor (the quantity by which you divide). Polynomials are algebraic expressions that involve sums or differences of several terms. Binomials are a specific type of polynomial that has exactly two terms, separated by a plus or minus sign.
02

Recognize Division Process

The division process consists of subtracting the divisor's product with the partial quotient from the dividend, or the part of the dividend that you've considered so far (starting from its highest-order term). The next term of the partial quotient is then the ratio of the highest-order term of the result to the highest-order term of the divisor.
03

Identify Stopping Condition

The division process finishes when the degree (highest power) of the remaining part of the dividend (after subtracting the divisor's product with the partial quotient) is less than the degree of the divisor. That's because a polynomial cannot be divided by another polynomial of higher degree and yield a term for the quotient. In this case, what remains is known as the remainder of the division process. It can sometimes be zero, which denotes that the original polynomial was exactly dividable by the divisor.

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{aligned}&3 y^{3}+2 y^{2}+y+4\\\&y+3\end{aligned}$$

Will help you prepare for the material covered in the next section. In each exercise, find the indicated products. Then, if possible, state a fast method for finding these products. (You may already be familiar with some of these methods from a high school algebra course.) a. \((x+3)(x-3)\) b. \((x+5)(x-5)\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Other than multiplying monomials, the distributive property is used to multiply other kinds of polynomials.

We have seen that in \(2009,\) the United States government spent more than it had collected in taxes, resulting in a budget deficit of \(\$ 1.35\) trillion. a. Express 1.35 trillion in scientific notation. b. A trip around the world at the Equator is approximately \(25,000\) miles. Express this number in scientific notation. c. Use your scientific notation answers from parts (a) and (b) to answer this question: How many times can you circle the world at the Equator by traveling 1.35 trillion miles?

Graph \(3 x-2 y=6\) using intercepts. (Section 3.2 Example 4 )
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