Polynomials

Question 1

Factor theorem

Answer 1

If p(a)=0 (where a is a real number) then x-a is a factor of p(x)

Question 1

Remainder theorem

Answer 1

When the polynomial P(x) is divided by x-c, then the remainder is P(c), where c is a real number.
(reverse of factor theorem)

Question 2

How to do synthetic division?

Answer 2

1. Given the expression to solve for, make it equal to zero and solve for x (that is what you are going to divide by)
2. Set up the top row of values by inputting the coefficients and the constant of the expression - make sure to replace any missing degrees with a zero
3. Move the leading coefficient to the botttom
4. Multiply this number by the divisor and move the product to the second column, second row
5. Add the top value with the product and bring the solution to the bottom
6. Repeat this process until all places have been filled
7. If the final solution does not equal zero, that is the remainder

Question 3

How to do long division?

Answer 3

1. Arrange the terms in the decreasing order of their indices (if required). Write the missing terms with zero as their coefficient.
2. For the first term of the quotient, divide the first term of the dividend by the first term of the divisor.
3. Multiply this term of the quotient by the divisor to get the product.
4. Subtract this product from the dividend, and bring down the next term (if any). The difference and the brought down term will form the new dividend.
5. Follow this process until you get a remainder, which can be zero or of a lower index than the divisor.