A polynomial is an algebraic expression that has constants, variables, and coefficients with a point where the value of the polynomial becomes zero as a whole.
Answer: The value of a in 2 R1 - R2 = 0 is 18 / 127.
Here's the step-by-step solution.
Explanation:
Let f[ x ] = ax3 + 3x2 - 3 and g[ x ] = 2x3 - 5x + a
Given that f[ x ] and g[ x ] when divided by x - 4 leaves the remainders R1 and R2 respectively.
By remainder theorem, substituting the value x = 4 in both f[ x ] and g[ x ], we get remainders.
For f[ 4 ] = ax3 + 3x2 - 3
= a × [ 4 ]3 + 3 × [ 4 ]2 - 3 = 64 a + 48 - 3
= 64 a + 45 = R1 [eq 1]
For g[ 4 ] = 2x3 - 5x + a
= 2 × [ 4 ]3 - 5 × [ 4 ] + a = 128 - 20 + a
= 108 + a = R2 [eq 2]
Given that 2R1 - R2 = 0
Therefore, from eq 1 and eq 2
2 [ 64 a + 45 ] - [ 108 + a ] = 0
⇒ 128 a + 90 - 108 - a = 0
⇒ 127 a - 18 = 0
⇒ 127 a = 18
⇒ a = 18 / 127
Thus, the value of a in 2 R1 - R2 = 0 is 18 / 127.
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Solution
Given polynomials
P[
x1 ]= ax3 +3x2 - 3 andp[x2 ]= 2x3 - 5x + a
It is also given that these two polynomials leave the same remainder when divided by [x - 4].
i.e., [x-4] is the zero of the polynomial so, x=4
Now put the value of 'x' in the polynomials,
As both the Eq. have the same remainder so,
p[x1 ]=p[x2 ]
⇒ a[43 ] + 3[42 ] - 3 = 2[43 ] - 5[4] + a
64a + 48-3 = 128 - 20 + a
64a - a = 108 - 45
63 a = 63
a = 1
Solve
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