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If 2 and 0 are the zeros of the polynmi...

If 2 and 0 are the zeros of the polynmial `f(x)=2x^(3)-5x^(2)+ax+b` then find the values of a and b.

A

a = -2 , b = 0

B

a = 2 , b = 1

C

a = 2 , b = 0

D

a=2 , b= -1

Text Solution

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The correct Answer is:
To find the values of \( a \) and \( b \) in the polynomial \( f(x) = 2x^3 - 5x^2 + ax + b \) given that the zeros are \( 2 \) and \( 0 \), we will follow these steps: ### Step 1: Use the fact that \( 0 \) is a zero of the polynomial. Since \( 0 \) is a zero, we can substitute \( x = 0 \) into the polynomial: \[ f(0) = 2(0)^3 - 5(0)^2 + a(0) + b = b \] Since \( f(0) = 0 \), we have: \[ b = 0 \] ### Step 2: Use the fact that \( 2 \) is a zero of the polynomial. Now, we substitute \( x = 2 \) into the polynomial: \[ f(2) = 2(2)^3 - 5(2)^2 + a(2) + b \] Calculating each term: \[ = 2(8) - 5(4) + 2a + b \] \[ = 16 - 20 + 2a + b \] \[ = -4 + 2a + b \] Since \( f(2) = 0 \), we set the equation to zero: \[ -4 + 2a + b = 0 \] ### Step 3: Substitute the value of \( b \) into the equation. We already found that \( b = 0 \). Substituting this into the equation gives us: \[ -4 + 2a + 0 = 0 \] This simplifies to: \[ -4 + 2a = 0 \] ### Step 4: Solve for \( a \). Adding \( 4 \) to both sides: \[ 2a = 4 \] Dividing both sides by \( 2 \): \[ a = 2 \] ### Conclusion The values of \( a \) and \( b \) are: \[ a = 2, \quad b = 0 \]
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