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# If (X – 2) is a Factor of the Expression 2x3 + Ax2 + Bx – 14 and When the Expression is Divided by (X – 3), It Leaves a Remainder 52, Find the Values of a and B - ICSE Class 10 - Mathematics

#### Question

If (x – 2) is a factor of the expression 2x3 + ax2 + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b

#### Solution

Since (x-2) is a factor of polynomial  2x^3+ax^2+bx-14, we have

2(2)^3+a(2)^2+b(2)-14=0

⇒ 16+4a+2b-14=0

⇒4a+2b+2=0

⇒2a+b+1=0

⇒2a+b=-1          .............(1)

On dividing by (x-3), the polynomial 2x^3+ax^2+bx-14 leaves remainder 52,

⇒2(3)^3+a(3)^2+b(3)-14=52

⇒54+9a+3b-14=52

⇒9a+3b+40=52

⇒9a+3b=12

⇒3a+b=4       ...............(2)

Subtracting (1) and (2), we get

a=5

substituting a =5 in (1), we get

2xx55+b=-1

⇒10+b=-1

⇒b=-11

Hence , a=5 and b=-11.

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Solution If (X – 2) is a Factor of the Expression 2x3 + Ax2 + Bx – 14 and When the Expression is Divided by (X – 3), It Leaves a Remainder 52, Find the Values of a and B Concept: Remainder Theorem.
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