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find the value s of.. a and b if 0 and 2 are the zeros of the polynomial p(x) =2xcube minus 5xsquare plus ax plus b find the third zero of the polynomial ​

Sagot :

Answer:

a = 2, b = 0 , x = [tex]\frac{1}{2}[/tex]

Step-by-step explanation:

Given that x = 0 and x = 2 are zeros of p(x) , then

p(0) = 0 and p(2) = 0

p(0) = 2(0)³ - 5(0)² + a(0) + b = 0 , that is

0 - 0+ 0 + b = 0, then b = 0

p(2) = 2(2)³ - 5(2)² + 2a + b = 0 , that is

16 - 20 + 2a + 0 = 0

- 4 + 2a = 0 ( add 4 to both sides )

2a = 4 ( divide both sides by 2 )

a = 2

Then

p(x) = 2x³ - 5x² + 2x ← factor out x from each term

      = x(2x² - 5x + 2)

      = x(2x - 1)(x - 2)

To find zeros equate p(x) to zero

x(2x - 1)(x - 2) = 0

Equate each factor to zero and solve for x

x = 0

x - 2 = 0 ⇒ x = 2

These are the 2 zeros already known (given)

2x - 1 = 0 ⇒ 2x = 1 ⇒ x = [tex]\frac{1}{2}[/tex] ← third zero of p(x)

       

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