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Class 9

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>>Polynomials

>>Remainder Theorem

>>The polynomial ax^3 + bx^2 + x - 6 has (

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The polynomial ax

3

+bx

2

+x−6 has (x+2) as a factor and leaves a remainder 4 when divided by (x−2). Find a and b.

This question has multiple correct options

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Solution

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Correct options are A) and B)

Let p(x)=ax

3

+bx

2

+x−6

Since (x+2) is a factor of p(x), then by Factor theorem p(−2)=0

⇒a(−2)

3

+b(−2)

2

+(−2)−6=0

⇒−8a+4b−8=0

⇒−2a+b=2 ...(i)

Also when p(x) is divided by (x-2) the remainder is 4, therefore by Remainder theorem p(2)=4

⇒a(2)

3

+b(2)

2

+2−6=4

⇒8a+4b+2−6=4

⇒8a+4b=8

⇒2a+b=2 ...(ii)

Adding equation (i) and (ii), we get

(−2a+b)+(2a+b)=2+2

⇒2b=4⇒b=2

Putting b=2 in (i), we get

−2a+2=2

⇒−2a=0⇒a=0

Hence, a=0 and b=2