Question

if 2x^3 + ax^2 + bx-2 has a factor (x+2) and leaves a remainder, find the values of a and b. using the values of a and b,
factorise the given polynomial completely.
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Answer 1

Answer:

If 2x³ + ax² + bx - 2 has a factor of (x+2) and leaves a remainders 0 and 7 when divided by (2x-3)

Find out

Value of a and b

Solution

★ Given polynomial

2x³ + ax² + bx - 2

✞ (x + 2) is a factor of given polynomial

➞ x + 2 = 0

➞ x = - 2

Remainder = 0

2x³ + ax² + bx - 2 = 0

✞ Put the value of x

➞ 2(-2)³ + a(-2)² + b(-2) - 2 = 0

➞ 2 × (-8) + 4a - 2b - 2 = 0

➞ -16 + 4a - 2b - 2 = 0

➞ 4a - 2b = 18

➞ 2(2a - b) = 18

➞ 2a - b = 9 ----(i)

$$\rule{200}3$$

✞ (2x - 3) is a factor of given polynomial

➞ 2x - 3 = 0

➞ x = 3/2

✞ Put the value of x

Remainder = 7

2x³ + ax² + bx - 2 = 7

➞ 2(3/2)³ + a(3/2)² + 3b/2 - 2 = 7

➞ 2 × 81/8 + 9a/4 + 3b/2 = 7 + 2

➞ 27/4 + 9a/4 + 3b/2 = 9

➞ 27 + 9a + 6b/4 = 9

➞ 27 + 9a + 6b = 9 × 4

➞ 9a + 6b = 36 - 27

➞ 3(3a + 2b) = 9

➞ 3a + 2b = 3 ----(ii)

$$\rule{200}3$$

★ Multiply (i) by 2 and (ii) by 1

4a - 2b = 18

3a + 2b = 3

★ Add both the equations

➞ 4a - 2b + 3a + 2b = 18 + 3

➞ 7a = 21

➞ a = 21/7 = 3

★ Putting the value of a in eqⁿ (ii)

➞ 3a + 2b = 3

➞ 3 × 3 + 2b = 3

➞ 9 + 2b = 3

➞ 2b = 3 - 9

➞ 2b = - 6

➞ b = -6/2 = - 3

Hence,

Required value of a = 3

Required value of b = - 3

$$\rule{200}3$$

Step-by-step explanation:

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Answer 2

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