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Answers
Answer:
9
Step-by-step explanation:
Given :
The expression ax⁴ + bx³ - x² + 2x + 3, when divided by x² + x - 2 , gives 4x + 3 as remainder. then, a + 4b is equal to
Solution :
We know that,
P(x) = q(x) × d(x) + r(x)
p(x) = divident,
q(x) = Quotient
d(x) = divisor
r(x) = remainder
By using remainder theorem,
x² + x - 2 = 0
⇒ x² - x + 2x - 2 = 0
⇒ x (x - 1) +2 (x - 1) = 0
⇒ (x - 1)(x + 2) = 0
⇒ for product of 2 numbers to be 0,
Atleast one of them must be 0 (or) both are zeroes,.
⇒ (x - 1) = 0 (or) (x + 2) = 0
⇒ x = 1 (or) x = -2
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ax⁴ + bx³ - x² + 2x + 3 = q(x) × d(x) + r(x)
⇒ ax⁴ + bx³ - x² + 2x + 3 - r(x) = q(x) × d(x)
⇒ ax⁴ + bx³ - x² + 2x + 3 - (4x + 3) = q(x) × d(x)
⇒ ax⁴ + bx³ - x² - 2x = q(x) × d(x)
If x = 1,
⇒ ax⁴ + bx³ - x² - 2x = 0
⇒ a(1)⁴ + b(1)³ - (1)² - 2(1) = 0
⇒ a + b - 1 - 2 - 0
⇒ a + b - 3 = 0 ⇒ a + b = 3 ...(i)
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If x = -2,
⇒ ax⁴ + bx³ - x² - 2x = 0
⇒ a(-2)⁴ + b(-2)³ - (-2)² - 2(-2) = 0
⇒ -2[ a(-2)³ + b(-2)² - (-2) - 2] = 0
⇒ a(-2)³ + b(-2)² - (-2) - 2 = 0
⇒ -8a + 4b + 2 - 2 = 0
⇒ -8a+ 4b - 0 = 0
⇒ 8a - 4b = 0
⇒ 4a - 2b = 0 ...(ii)
By adding (i) × 2 and (ii),
We get,
⇒ 2 × (a + b) + (4a - 2b) = 2 × (3) + 0
⇒ 2a - 2b + 4a + 2b = 6
⇒ 6a = 6
⇒ a = 1
Substituting value of a in (i),
We get,
a + b = 3
⇒ 1+ b = 3 ⇒ b = 3 - 1
⇒ b = 2..
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⇒ a + 4b = 1 + 4 × (2) = 1 + 8 = 9