Synthetic division is a mathematical method used to divide polynomials in algebraic equations. This method is useful in simplifying long division procedures and reducing the complexity of algebraic expressions. It is commonly used in solving equations in algebra, calculus, and engineering.

In synthetic division, a polynomial is divided by a linear factor. The method involves encoding the coefficients of the polynomial in a table and performing a series of arithmetic operations to obtain the quotient and remainder. The synthetic division method can be used to find factors and roots of polynomials, to evaluate functions, and to solve equations. In this article, we will explain the steps involved in synthetic division and provide some examples to illustrate the method.

The first step in synthetic division is to write the polynomial in standard form, with its terms arranged in descending order of degree. For example, consider the polynomial P(x) = 3x^3 – 4x^2 + 2x – 5. This polynomial can be written as P(x) = 3x^3 + 0x^2 – 4x^2 + 2x – 5.

Next, we need to identify the linear factor by which we want to divide the polynomial. This factor is usually in the form of x – a, where a is a constant or a root of the polynomial. For example, if we want to divide P(x) by x – 2, we write the factor as x – 2 = 0 and solve for x, which gives us x = 2. This means that 2 is a root or a factor of the polynomial P(x).

Now we can construct a table for the synthetic division method using the coefficients of the polynomial P(x). The first row of the table consists of the coefficients of the polynomial, starting with the coefficient of the highest degree term. The second row consists of the same coefficients, except the first coefficient is replaced by zero. This is done to align the columns of the table with the powers of x corresponding to each term. The third row is the result of multiplying the divisor (x – 2) with the first term of the polynomial, which is 3x^3. This result is written in the second column of the table, starting from the second row. The next step is to add the first and second row coefficients to get the third row coefficients. This process is repeated until we get the last row, which corresponds to the quotient of the division. The final value in the last row is the remainder.

Here is a table showing the synthetic division of P(x) by x – 2:

3 -4 2 -5
2 | 3 0 -4 2 -5
6 12 16
3 6 -2 18
6 4 16
10 30

The final row of the table represents the coefficients of the quotient polynomial, which is Q(x) = 3x^2 + 6x – 2. The remainder is 10. Therefore, we can write the polynomial P(x) as P(x) = (x – 2)Q(x) + R, where R is the remainder. In other words, we have factored P(x) into the product of (x – 2) and Q(x), and we have also found the remainder when P(x) is divided by (x – 2).

Here are some other examples of synthetic division:

Example 1: Divide the polynomial P(x) = 4x^3 – 3x^2 + 2x + 1 by x + 2.

4 -3 2 1
-2 | 4 -8 22 -46
4 -11 11
-7 13
-25

The quotient is Q(x) = 4x^2 – 11x + 13, and the remainder is -25.

Example 2: Divide the polynomial P(x) = x^4 + 2x^3 – 5x^2 – 7x + 2 by x – 3.

1 2 -5 -7 2
3 | 1 5 10 5 -32
3 24 45 144
1 8 15 49
4 7 64
-5 -121

The quotient is Q(x) = x^3 + 5x^2 + 10x + 49, and the remainder is -5.

In conclusion, synthetic division is a useful method for dividing polynomials by linear factors. It simplifies the long division process and allows us to factorize and solve equations more easily. By following the steps outlined in this article and practicing with more examples, you can become proficient in using synthetic division to solve algebraic problems.

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