5.5 Apply the Remainder and Factor Theorems

Divisor Quotient Remainder

When you divide a polynomial f(x) by a divisor d(x) you get a quotient polynomial q(x) and a remainder polynomial r(x). Remainder Theorem: If a polynomial f(x) is divided by x – k Then the remainder is r = f(k)

Recall to evaluate a polynomial you could use direct substitution. For example evaluate: when x = 3. And…by synthetic substitution: 2 0 -8 5 -7 3 x-value ADD 105 6 18 30 10 35 98 6 Mult. By 3 2 f(3)=98

2 0 -8 5 -7 6 18 30 105 3 2 6 10 35 98 This is equivalent to dividing By x - 3 synthetic substitution: 2 0 -8 5 -7 6 18 30 105 3 2 6 x-value 10 35 98 or

Use synthetic division to divide: By x - 2

Factor Theorem: A polynomial f(x) has a factor x – k if and only if f(k)=0. Factor given that f(-3) = 0. Because f(-3) = 0, you know that x – (-3) or x+3 is a factor of f(x). Use synthetic division to find the other factors: Solution: -3 2 11 18 9 -6 -15 -9 2 5 3 0