Answer:
1.) a = 73, b = 60
2.) H(0) = -60
Explanation:
Since equation 2x2 - 7x - 15 is a factor of H(x) = 6x3 - 13x2 - ax - b where a and b are constants.
Find the root of 2x^2 - 7x - 15
2x^2 - 10x + 3x - 15 = 0
2x( x - 5 ) + 3( x - 5 ) = 0
2x + 3 = 0 or x - 5 = 0
2x = -3 or x = 5
X = -3/2 or 5
Substitute the two values into the H(x)
When x = - 3/2
6(-3/2)^3 - 13( -3/2 )^2 - a( -3/2 ) - b = 0
6(-27/8) - 13( 9/4 ) + 3a/2 -b = 0
-81/4 - 117/4 + 3a/2 - b = 0
- 198 + 6a - 4b = 0 .... (1)
When x = 5
6( 5 )^3 - 13( 5 )^2 - a(5) - b = 0
6(125) - 13(25) - 5a - b = 0
750 - 325 - 5a - b = 0
425 - 5a - b = 0 ..... ( 2)
Solve equation 1 and 2 simultaneously
Eliminate b
- 198 + 6a - 4b = 0 × 1
425 - 5a - b = 0 × 4
- 198 + 6a - 4b = 0
1700 - 20a - 4b = 0
- 1898 + 26a = 0
26a = 1898
a = 1898/26
a = 73
Substitute a back into equation 1
- 198 + 6(73) - 4b = 0
-198 + 438 - 4b = 0
4b = 240
b = 240/4
b = 60
2.) To Find the zeros of H(X), substitute 0 for x. that is,
H(0) = 6(0)^3 - 13(0)^2 - 73(0) - 60
H(0) = - 60