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 2 + 5 x - 7 when x = 3.

REMAINDER THEOREM

Polynomial in standard form

2 x 4 + 0 x

3 – 8 x 2 + 5 x – 7

2 6

6

10

18

35

30 105

98

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

2 0 –8 5 –7 CoefficientsCoefficients

3

x-value

3 •

SOLUTION

Polynomial instandard form

Using direct substitution to evaluate polynomial functions is another alternative, lets compare.

Evaluate

f (x) = 2 x 4 -8 x

2 + 5 x - 7 when x = 3.

f(x)=2x4-8x2+5x-7Find f(3)f(3)=2(3)4-8(3)2+5(3)-7f(3)= 2(81)-8(9)+15-7f(3)=162-72+15-7f(3)=98

Use synthetic substitution

f (x) = 3 x 4 -2 x

3 + x

2 - 2 find f(4)

Polynomial in standard form

3 x 4 – 2 x

3 + x 2 + 0 x – 2

3 10

12

41

40

164

164 656

654

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)=654

3 -2 1 0 –2 CoefficientsCoefficients

4

x-value

4 •

SOLUTION

Polynomial instandard form

Use synthetic substitution

3 24 15 18, fi nd 3f x x x x f

SOLUTION 3 1 4 -15 -18

3 21 18

1 7 6 0

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

3 2 4 15 18,

given that 3 is a zero of ,

determine all other zeros and write

the polynomial in terms of a product of

linear and/ or irreducible quadratic f actors

f x x x x

f x

SOLUTION 3 1 4 -15 -18

3 21 18

1 7 6 0

23 7 6f x x x x

3 6 1f x x x x

RATIONAL ZERO THEOREM

pq

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. 3 22 5 4 12p x x x x

Identify p and q p=1, 2, 3, 4, 6, 12

q=1, 2

Find pq

1 2 3 4 6 12 1 2 3 4 6 12, , , , , , , , , , ,

1 1 1 1 1 1 2 2 2 2 2 2

1 31, 2, 3, 4, 6, 12, ,

2 2 Simplify and eliminate

duplicates.

HOMEWORK

Pages 355-356 55-71 EOO, 78-92 ALL379-381; 1-19 ODD, 25, 77-79 ALL