Exercise 2 Page 177

Synthetic division is used to divide a polynomial by a binomial in a concise way. This requires us to follow certain conventions. In particular, the divisor must be of the form x-k. This reverses the sign of the k-value, which allows us to add instead of subtracting in the division algorithm. Let's see how to ensure this for different cases.

Dividing by a binomial of the form (x+a)

In these cases, we can rewrite the binomial as x-(- a) and use - a as our k-value. Example [0.8em] (2x^3-x^2-4) ÷ (x+3) ⇕ (2x^3-x^2-4) ÷ (x- ( - 3))Therefore, to divide by x+3 we divide by x-(- 3) and use the k-value k= - 3.

Dividing by a binomial of the form bx-a

Adapting this binomial requires a bit of work. First, we can rewrite the binomial as b(x- ab). Then, to perform the division by b(x- ab) we can divide all the dividend coefficients by b first. Finally, we divide by x- ab using synthetic division with k= ab as our k-value. Example 4x^3-2x^2-8 ÷ 2x-1 First, let's start by factoring out the coefficient of the x-term — in this case, 2. Then we can divide out this factor.

Now we we can rewrite our original division. 4x^3-2x^2-8 ÷ 2x-1 ⇕ 2x^3-x^2-4 ÷ x- 12 In this new format, we can use synthetic division by using k = 1 2 as the k-value.

Conclusion

The conventions for using the algorithm of synthetic division requires the divisor to be of the form x-k. Considering the cases mentioned above, as long as the binomial has the form ax+b we can rewrite our division so that the divisor take the form x-k. Then we proceed as usual by using the synthetic division algorithm.