뒤로Dividing Polynomials: Long Division, Synthetic Division, Remainder and Factor Theorems
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Polynomial and Rational Functions
Dividing Polynomials: Long Division, Synthetic Division, Remainder and Factor Theorems
This section covers essential techniques for dividing polynomials, including long division and synthetic division. It also introduces the Remainder Theorem and Factor Theorem, which are fundamental for solving polynomial equations and evaluating polynomial functions.
Long Division of Polynomials
Long division is a systematic method for dividing one polynomial by another, similar to numerical long division.
Step 1: Arrange the terms of both the dividend and the divisor in descending powers of the variable.
Step 2: Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient.
Step 3: Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend, aligning like terms.
Step 4: Subtract the product from the dividend.
Step 5: Bring down the next term in the original dividend and write it next to the remainder to form a new dividend.
Step 6: Repeat the process until the degree of the remainder is less than the degree of the divisor.
Example: Divide by using long division.
Division Algorithm: If and are polynomials, then there exist unique polynomials and such that:
The remainder is either 0 or has a degree less than . If , divides evenly.
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form . It is especially useful for polynomials with missing terms.
Step 1: Arrange the polynomial in descending powers, using 0 for any missing term.
Step 2: Write for the divisor and list the coefficients of the dividend.
Step 3: Write the leading coefficient of the dividend on the bottom row.
Step 4: Multiply by the value just written and place the product in the next column.
Step 5: Add the values in the column and write the sum in the bottom row.
Step 6: Repeat until all columns are filled.
Step 7: The numbers in the last row give the coefficients of the quotient and the remainder.
Example: Use synthetic division to divide by . Here, and the coefficients are .
The Remainder Theorem
The Remainder Theorem states that when a polynomial is divided by , the remainder is .
Application: To evaluate , perform synthetic division with and the coefficients of . The remainder is .
Example: Given , find using the Remainder Theorem. The remainder after synthetic division is , so .
The Factor Theorem
The Factor Theorem connects zeros of a polynomial to its factors:
If , then is a factor of .
If is a factor of , then .
Example: If has a zero at , then is a factor of . Use synthetic division to factor $f(x)$ further.
Summary Table: Division Methods and Theorems
Method/Theorem | Purpose | Key Steps | Example |
|---|---|---|---|
Long Division | Divide any polynomial by another | Arrange, divide, multiply, subtract, repeat | by |
Synthetic Division | Divide by binomial | List coefficients, multiply, add, repeat | by |
Remainder Theorem | Evaluate | Divide by , remainder is | for |
Factor Theorem | Find factors and zeros | If , is a factor | is a factor if |
