Question

Let R1, and R2, be the remainders when the polynomials f(x)= x³ + 2ax² - 5x - 7 and g(x) = x³+ x²-12x+6a are divided by (x + 1) and (x - 2) respectively. If 2R¹+ R²= 12, find the value of 'a'.​

Answer 1

Answer:

5a-6

Step-by-step explanation:

Let p(x)=x

3 +2x 2−5ax−7

and q(x)=x³+ax 2 −12x+6 be the given polynomials

Now, R 1= Remainder when p(x) is divided by x+1.

⇒R

1

=p(−1)

⇒R

1

=(−1)

3

+2(−1)

2

−5a(−1)−7[∵p(x)=x

2

+2x 2 −5ax−7]

And R

2

= Remainder when q(x) is divided by x-2

⇒R

1

=q(2)

⇒R

2

=(2)

3

+a×2

2

−12×2+6[∵q(x)=x

2

+ax

2

−12x−6]

⇒R

2

=8+4a−24+6

⇒R

2

=4a−10

Substituting the values of R

1

and R

2

in 2R

1

+R

2

=6, we get

⇒2(5a−6)+(4a−10)=6

⇒10a−12+4a−10=6

⇒14a−22=6

⇒14a−28=0

⇒a=2

⇒R 1=−1+2+5a−7

⇒R1=5a−6