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Synthetic Division Steps (synthetic division method of polynomials)-
Synthetic Division of Polynomials : The synthetic division method of polynomials is a way of dividing one polynomial by another to find the quotient and the remainder. It uses the same principles as long division, but it is a shorter and easier process.
Synthetic Division of Polynomials Steps
To divide a polynomial by another using synthetic division, you first need to write the division symbol (÷) and the dividend (the polynomial that is being divided) above the divisor (the polynomial that is doing the dividing).
Then, you need to list the coefficients of the dividend (the numbers that appear in front of the variables) and the coefficients of the divisor (the numbers that appear in front of the variables in the divisor) in descending order.
Next, you need to draw a line under the coefficients of the dividend and a line under the coefficients of the divisor.
Then, you need to draw a diagonal line from the upper left-hand corner of the dividend to the upper left-hand corner of the divisor.
Next, you need to start dividing the coefficients of the dividend by the coefficients of the divisor.
- For the first coefficient, you divide the dividend’s coefficient by the divisor’s coefficient. The answer is the quotient.
- For the second coefficient, you multiply the dividend’s coefficient by the quotient and add the divisor’s coefficient. This is the new quotient.
Synthetic Division of Polynomials Examples
In mathematics, synthetic division is a method for dividing a polynomial by a linear factor, x − a. Synthetic division can be used for polynomials with real or complex coefficients and for polynomials whose degree is greater than 1. The method is also sometimes called long division of polynomials. The goal of synthetic division is to simplify the division process and to avoid having to factor the polynomial. The method is based on the fact that if x − a is a factor of a polynomial P(x), then P(a) = 0.
To divide a polynomial P(x) by x − a, synthetic division can used. The steps for synthetic division are as follows:
1. Write the polynomial P(x) in standard form, with the term of highest degree first.
2. Write the number a underneath the leading coefficient of P(x).
3. Draw a line under the terms of P(x).
4. Multiply each term of P(x) by a, and write the result underneath the term.
5. Add the terms in the lower row, and write the result underneath the line. This sum is P(a).
6. If P(a) = 0, then x − a is a factor of P(x). Otherwise, x − a is not a factor of P(x).
For example, to divide x^3+4x^2-5x-6 by x+2, synthetic division can used. The steps are as follows:
1. Write the polynomial in standard form: x^3+4x^2-5x-6.
2. Write the number 2 underneath the leading coefficient: 2.
3. Draw a line under the terms of the polynomial: x^3+4x^2-5x-6.
4. Multiply each term of the polynomial by 2, and write the result underneath the term: 2x^3+8x^2-10x-12.
5. Add the terms in the lower row, and write the result underneath the line: 2x^3+8x^2-10x-12. This sum is P(2), which is 0.
