Synthetic Division of Polynomials

Front

What is synthetic division?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It is faster than long division and involves using the coefficients of the polynomial.

Front

What is the first step in synthetic division?

Back

The first step is to write down the coefficients of the polynomial in descending order of degree, filling in any missing degrees with zeros.

Front

How do you set up synthetic division for (2x^3 + 5x^2 + 9) ÷ (x + 3)?

Back

Set up synthetic division by using -3 (the root of x + 3) and the coefficients: 2, 5, 0, 9.

Front

What is the result of synthetic division for (3x^3 - 2x^2 - 7x + 6) ÷ (x + 1)?

Back

The result is 3x^2 - 5x - 2 with a remainder of 8.

Front

What do you do with the remainder in synthetic division?

Back

The remainder is added to the quotient as a fraction over the divisor. For example, if the remainder is r and the divisor is (x - c), the result is quotient + r/(x - c).

Front

How do you determine the number of zeros needed in synthetic division?

Back

The number of zeros needed corresponds to the highest degree of the polynomial minus the degree of the divisor.

Front

What is the significance of the coefficients in synthetic division?

Back

The coefficients represent the values of the polynomial at different degrees and are used to perform the division process.

Front

What is the final answer format for synthetic division?

Back

The final answer is written as the quotient plus the remainder over the divisor, e.g., quotient + remainder/(x - c).

Front

How do you handle missing degrees in a polynomial during synthetic division?

Back

You replace any missing degrees with a coefficient of zero to maintain the correct order.

Front

What is the quotient when dividing (x^4 + 2x^3 - 3x + 5) by (x - 1)?

Back

The quotient is x^3 + 3x^2 + 3x + 0 with a remainder of 8.