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Chapter 5: Problem 52

When dividing a binomial into a polynomial with missing terms, explain the advantage of writing the missing terms with zero coefficients.

Short Answer

Expert verified
Writing the missing terms with zero coefficients when dividing a binomial into a polynomial is advantageous. It ensures that every term of the polynomial is accounted for, thereby making the dividing process clear and less complicated. This method doesn't alter the value of the polynomial but helps to maintain the structure during the operation.

Step by step solution

01

Understanding the terms

It is important to understand what the terms binomial and polynomial mean. A binomial is an algebraic expression containing two terms, while a polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
02

Simulate a division operation

Let us simulate an operation: Divide \(x^3 - 2x^2 + 3\) by \(x-2\). In this polynomial, no coefficient is missing, hence it's quite straightforward to divide it using long division or synthetic division.
03

Identifying missing terms

Now observe a similar operation: Divide \(x^3 - 2x + 3\) by \(x-2\). Here, \(x^2\) term is missing in the dividend polynomial. It would be difficult and confusing to proceed with the division without a place value for \(x^2\).
04

rewrite with zero coefficients

The polynomial can be rewritten as: \(x^3 + 0x^2 - 2x + 3\). Now, the polynomial is clear, and division is straightforward. The zero coefficient doesn鈥檛 affect the value of the polynomial but helps to maintain the structure of the polynomial, which would ease the division process.

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