Theory of equations/Polynomial division and the Remainder Theorem

The topic "Polynomial division and the Remainder Theorem" introduces the algebraic machinery that underlies much of the elementary theory of equations. In this chapter we describe how to divide one polynomial by another, how to interpret the remainder of such a division, and how this leads to efficient methods for evaluating polynomials and detecting factors. Polynomial division in one variable Polynomial division is the analogue for polynomials of the long division algorithm for integers. Throughout this section we work over a field such as the real numbers or complex numbers; however, the same formal procedure applies over any commutative ring. Setup and notation Let and be polynomials in one indeterminate , with . We write where , and . The goal of polynomial division is to find polynomials (the quotient) and (the remainder) such that with either or . The division algorithm for polynomials The polynomial division algorithm proceeds as follows: Arrange both and in descending powers of , inserting zero coefficients as needed. Compare the leading terms of and of . Form the term and add this term to the quotient . Subtract from to obtain a new polynomial of strictly smaller degree. Replace by this new polynomial and repeat the process until the degree of the new is strictly smaller than . This final is the remainder . The polynomial division algorithm always terminates, and the quotient and remainder are unique. Worked example: dividing by a linear polynomial Consider and divide by Leading terms: and . Their quotient is . Begin with . Multiply and subtract: Now divide by . Leading terms: and . Their quotient is , so add to the quotient: . Multiply and subtract: Now divide by . Leading terms: and . Their quotient is . Update the quotient: . Multiply and subtract: The remainder is the constant polynomial . We have This example illustrates the general pattern of polynomial long division. Synthetic division For division by a linear polynomial of the form , there is a more efficient variant known as synthetic division. Synthetic division simplifies computations by working only with coefficients. Procedure for synthetic division Suppose we wish to divide by . The steps are: Write down the coefficients in a row. Bring down as the first coefficient of the quotient. Multiply the current number by and write the product underneath the next coefficient. Add vertically to obtain the next coefficient of the quotient. Repeat this multiply–add process across the row of coefficients. The final number obtained (after the last addition) is the remainder, while the previous numbers form the coefficients of the quotient polynomial. Example: synthetic division by To divide by using synthetic division: Coefficients of : . Divisor root: . We can display the steps using a wikitable: Step Operation Result 1 Bring down the leading coefficient 2 Multiply by and add to next coefficient , 3 Multiply by and add to next coefficient , 4 Multiply by and add to last coefficient , So the final row of numbers is: . Thus the quotient and remainder are: Quotient: . Remainder: . This agrees with the result of long division above. The Remainder Theorem The Remainder Theorem explains why synthetic division works and gives an immediate way to evaluate polynomials at a point. Statement of the theorem Remainder Theorem. Let be a polynomial and let be any number. When we divide by the linear polynomial , the remainder is equal to . More precisely, there exists a polynomial and a constant such that and the Remainder Theorem asserts that . Proof Begin with the division algorithm applied to and . There exist a polynomial and a constant such that Substitute into this identity: Hence the remainder is exactly , as claimed. Example: applying the Remainder Theorem Consider and the divisor . Direct evaluation: So the remainder should be . Dividing by using the algorithm or synthetic division indeed produces remainder . This confirms the theorem in this instance. The Factor Theorem The Factor Theorem is an immediate consequence of the Remainder Theorem and connects zeros of polynomials with linear factors. Statement of the theorem Factor Theorem. Let be a polynomial and let be a number. Then is a factor of if and only if . In other words, Proof Suppose is a factor of . Then there exists a polynomial such that Substituting gives Conversely, suppose . By the division algorithm for polynomials there exist a polynomial and a constant such that By the Remainder Theorem, the remainder is . Thus so divides . Example: factoring a cubic polynomial Factor the cubic We look for integer roots by evaluating at small integers: Each of is a root, so by the Factor Theorem, , , and are factors. We can write This factorization can be confirmed by multiplying out the three linear factors. Using polynomial division to factor polynomials Polynomial division and the Factor Theorem together provide an efficient strategy for factoring polynomials of moderate degree. Basic strategy Given a polynomial with integer coefficients: Use the Rational Root Test (if applicable) to list candidate rational roots based on the factors of the constant term and the leading coefficient. Evaluate at these candidate values using direct substitution or synthetic division. If for some candidate , then is a factor of . Use polynomial long division or synthetic division to divide by and obtain a quotient of lower degree. Repeat the process on until the polynomial is fully factored or no further rational roots are found. Example: factoring with repeated roots Factor Step 1. Factor out any common factors: Step 2. Look for roots of the cubic . Evaluating at small integers: Hence is a factor. Divide the cubic by . Using synthetic division with coefficients and : Step Operation Result 1 Bring down the leading coefficient 2 Multiply by and add to next coefficient , 3 Multiply by and add to next coefficient , 4 Multiply by and add to last coefficient , The quotient is , and the remainder is . So Combining with the factor extracted earlier, the full factorization is Applications to the theory of equations Polynomial division and the Remainder Theorem occupy a central position in the theory of algebraic equations: They provide effective methods for evaluating polynomials at numeric values, especially via synthetic division. They allow one to extract linear factors corresponding to known roots, thereby reducing higher-degree equations to lower-degree ones. In combination with other techniques (such as partial fraction decomposition, root localization, and numerical approximation), they support more advanced analysis of polynomial equations. In subsequent chapters, these tools will be used to study: the structure of polynomial equations of degree 2, 3, and 4, relationships among roots and coefficients, and methods for approximating roots when explicit algebraic solutions are not available.