5.5C-Remainder & Factor Theorems

5.5C-Remainder & Factor Theorems • BELL RINGERS • 1.) What are the SOLUTIONS of (x-2)(X+3) • 2.) What are the FACTORS if the solutions are 0, 7 and -4 ? • 3.) Find the FACTORS and SOLUTIONS of • A) • B)

Remainder & Factor Theorems • Remainder Theorem: If a polynomial is divided by (x –k), then the REMAINDER is given by r=f(k); remainder = f(k) • Factor Theorem: If remainder = 0 (f(k)=0) then x-k is a GOOD FACTOR and k is a GOOD SOLUTION

Finding ALL solutions • 1.) Synthetically divide by k (oppositeby “a” of factor) • 2.) Write Quotient (start with x ONE POWER SMALLER than original) • 3. If Remainder = 0, SOLVE QUOTIENT to find the rest of the solutions • A.) factor • B.) quadratic formula • C.) get alone (NO ), then square root • D.) get x alone ( quotient linear with degree=1: x)

Examples: • 1. Factor completely & solve. One GOOD factor is (x-4). F(x)=

Example: Factor Completely & Solve • 2. f(x) = ; x-6

Example: Given f(x) and ONE ZERO, find the other zeros • 3. f(x)=-18 one solution is -3

Example: Find ALL solutions • 4. f(x)= f(-2)=0

Example: Find an expression for the missing dimensions • 5. V= X-1 X+6 ?

5.5C-Remainder & Factor Theorems