MHF4U - Unit 2 - Polynomial Equations and Inequalities

When a polynomial is divided by a binomial, there may be a remainder
To divide a polynomial by a binomial
- Write out the polynomial, giving each degree its own column. If a degree is not present in the polynomial, give it a coefficient of 0
- Find the factor that turns the term in the binomial with a variable into the first term in the polynomial
- Write this factor above its appropriate degree column
- Distribute this factor to both values in the binomial, and write this product under the polynomial in its given degree columns
- Subtract
- Repeat until there are no terms left in the initial polynomial
The Remainder Theorem - If a polynomial P(x) is divided by x - b, the remainder is the constant R
- Quotient form - [p(x)]/(x - b) = Q(x) + R/(x - b)
  - Where Q(x) is a polynomial with a degree one less than the degree of P(x)
  - If the remainder is 0, then it is just Q(x)
- P(b) is equal to the polynomial divided by (x - b)

The Factor Theorem - The binomial (x-b) is a factor of P(x) if and only if R = 0
- P(b) = 0, b is a root of the function, x - b is a factor
The Rational Zero Theorem - The reduced fraction b/a is a possible rational zero, when “b” is a factor of the constant term and “a” is a factor of the leading coefficient
- List all (positive and negative) factors of the constant term
- List all (positive and negative) factors of the leading coefficient
- List all the possible rational zeros or roots (as a list of b/a)
To find an actual zero, substitute these possible roots into the initial equation
- When x=(b/a) and P(x)=0, x-(b/a) is a factor
- Divide P(x) by this factor to get another polynomial
- Continue until the quotient is a quadratic polynomial, where it can be factored normally
In some cases, the zeros can also be found by factoring using perfect square trinomial or difference of squares