## Presentation ~ above theme: "If a polynomial f(x) is divided by (x-a), the remainder (a constant) is the worth of the duty when x is equal to a, i.e. F(a). Therefore, we deserve to use."— Presentation transcript:

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1 If a polynomial f(x) is divided by (x-a), the remainder (a constant) is the value of the function when x is same to a, i.e. F(a). Therefore, we can use synthetic division to assist us evaluate functions through a process called synthetic substitution. Evaluate f (x) = 2 x 4 8 x 2 + 5 x 7 as soon as x = 3. REMAINDER theorem

2 Polynomial in standard type 2 x 4 + 0 x 3 – 8 x 2 + 5 x – 7 2 6 6 10 18 35 30105 98 The value of f (3) is the last number friend write, In the bottom right-hand corner. Below f(3)=98 The worth of f (3) is the critical 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 type

3 Using straight substitution to advice polynomial features is one more alternative, lets compare. Advice f (x) = 2 x 4 8 x 2 + 5 x 7 as soon as 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)=162-72+15-7 f(3)=98

4 Use synthetic substitution f (x) = 3 x 4 x 3 + x 2 uncover f(4)

5 Polynomial in standard form 3 x 4 – 2 x 3 + x 2 + 0 x – 2 3 10 12 41 40 164 656 654 The worth of f (4) is the critical number friend write, In the bottom right-hand corner. Right here f(4)=654 The value of f (4) is the last number friend write, In the bottom right-hand corner. Below f(4)=654 3-210 –2 Coefficients 4 x -value 4 S OLUTION Polynomial in standard form

7 factor Theorem If P(a)=0, climate x-a is a element of P(x). Whereas if x-a is a element of P(x), then P(a)=0

9 rational ZERO theorem If a polynomial duty has essence coefficients, then every rational zero of P(x) has actually the form where p are the factors of the consistent and q space the components of the top coefficient

10 reasonable ZERO THEOREM usage the reasonable zero organize to perform the possible rational zeros.

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Recognize p and q p=1, 2, 3, 4, 6, 12 q=1, 2 Simplify and eliminate duplicates.

11 HOMEWORK Pages 355-356 55-71 EOO, 78-92 all 379-381; 1-19 ODD, 25, 77-79 every