If a polynomial f(x) is divided by (x-a), the remainder (a constant) is the value of the function when x is equal to a, i.e. f(a). Therefore, we can use synthetic division to help us evaluate functions through a process called synthetic substitution. Evaluate f (x) = 2 x 4  8 x x  7 when x = 3. REMAINDER THEOREM

Polynomial in standard form 2 x x 3 – 8 x x – The value of f (3) is the last number you write, In the bottom right-hand corner. Here f(3)=98 The value of f (3) is the last number you write, In the bottom right-hand corner. Here f(3)=98 20–85 –720–85 –7 Coefficients 3 x -value 3 S OLUTION Polynomial in standard form

Using direct substitution to evaluate polynomial functions is another alternative, lets compare. Evaluate f (x) = 2 x 4  8 x x  7 when x = 3. f(x)=2x 4 -8x 2 +5x-7 Find f(3) f(3)=2(3) 4 -8(3) 2 +5(3)-7 f(3)= 2(81)-8(9)+15-7 f(3)= f(3)=98

Use synthetic substitution f (x) = 3 x 4  x 3 + x 2  find f(4)

Polynomial in standard form 3 x 4 – 2 x 3 + x x – The value of f (4) is the last number you write, In the bottom right-hand corner. Here f(4)=654 The value of f (4) is the last number you write, In the bottom right-hand corner. Here f(4)= –2 Coefficients 4 x -value 4 S OLUTION Polynomial in standard form

Use synthetic substitution S OLUTION f(3)=0, what does that mean? Two very important concepts. 1.3 is a zero of the function. 2.x-3 is a factor of the polynomial.

Factor Theorem If P(a)=0, then x-a is a factor of P(x). Conversely, if x-a is a factor of P(x), then P(a)=0

S OLUTION

RATIONAL ZERO THEOREM If a polynomial function has integer coefficients, then every rational zero of P(x) has the form where p are the factors of the constant and q are the factors of the leading coefficient

RATIONAL ZERO THEOREM Use the rational zero theorem to list the POSSIBLE rational zeros. Identify p and q p=1, 2, 3, 4, 6, 12 q=1, 2 Simplify and eliminate duplicates.

HOMEWORK Pages EOO, ALL ; 1-19 ODD, 25, ALL