Question

the polynomial 2 x power 3 + A x power 2 + 3 x minus 5 and x power 3 + x power 2 - 2 X + a leaves the same remainder when divided by x minus 2 find the value of a also find the remainder in each case​

Answer 1

Answer:

The Remainder Theorem states that when you divide a polynomial p(x) by any factor (x−a); which is not necessarily a factor of the polynomial; you will obtain a new smaller polynomial and a remainder, and this remainder is the value of p(x) at x=a, that is p(a).

Let p(x)=2x

3

+ax

2

+3x−5 and q(x)=x

3

+x

2

−4x−a and the factor given is g(x)=x−1, therefore, by remainder theorem, the remainders are p(1) and q(1) respectively and thus,

p(1)=(2×1

3

)+(a×1

2

)+(3×1)−5=(2×1)+(a×1)+3−5=2+a−2=a

q(1)=1

3

+1

2

−(4×1)−a=1+1−4−a=−2−a

Now since it is given that both the polynomials p(x)=2x

3

+ax

2

+3x−5 and q(x)=x

3

+x

2

−4x−a leave the same remainder when divided by (x−1), therefore p(1)=q(1) that is:

a=−a−2

⇒a+a=−2

⇒2a=−2

⇒a=−

2

2

⇒a=−1

Hence, a=−1.

Step-by-step explanation:

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