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प्रश्न
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.
If (x – 2) is a factor of 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.
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उत्तर
Since (x – 2) is a factor of polynomial 2x3 + ax2 + bx – 14, we have
2(2)3 + a(2)2 + b(2) – 14 = 0
⇒ 16 + 4a + 2b – 14 = 0
⇒ 4a + 2b + 2 = 0
Dividing the entire equation by 2,
⇒ 2a + b = –1 ...(1)
On dividing by (x – 3), the polynomial 2x3 + ax2 + bx – 14 leaves remainder 52,
2(3)3 + a(3)2 + b(3) – 14 = 52
⇒ 54 + 9a + 3b – 14 = 52
⇒ 9a + 3b = 52 – 40
⇒ 9a + 3b = 12
Dividing the entire equation by 3,
⇒ 3a + b = 4 ...(2)
Subtracting (1) and (2), we get
2a + b = –1
3a + b = 4
– – –
–a = –5
Substituting a = 5 in (1), we get
2 × 5 + b = –1
⇒ 10 + b = –1
⇒ b = –11
Hence, a = 5 and b = –11.
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