Question

Polynomial x³ - ax² + bx - 6 leaves remainder -8 when divided by x - 1 and x - 2 is a factor of it.

Find the values of 'a' and 'b'.​

Answer 1

Step-by-step explanation:

x - 2 = 0

x = 2

$p(x) = {x}^{3} - a {x}^{2} + bx - 6 \\ p(2) = {(2)}^{3} - a {(2)}^{2} + b(2) - 6 \\ 0= 8 - 4a + 2b - 6 \\0 = 2 - 4a + 2b \\ 0 = 2(1 - 2a + b) \\ 0 = 1 - 2a + b \\ - 1 = - 2a + b \\ 2a - b = 1 \\ 2a = 1 + b \\ b = 2a - 1...(1)$

x - 1 = 0

x = 1

when p(x) is divided by x - 1, the remainder is (-8)

.°. p(1) = (-8)

$p(x) = {x}^{ 3 } - a {x}^{2} + bx - 6 \\ p(1) = {(1)}^{ 3 } - a {(1)}^{2} + b(1) - 6 \\ - 8 = 1 - a + b - 6 \\ - 8 = - 5 - a + b \\ - 8 + 5 = - a + b \\ - 3 = - a + b \\ a - b = 3 \\ a - (2a - 1) = 3 \\ a - 2a + 1 = 3 \\ - a = 2 \\ a = (- 2)$

Substituting a = (-2) in (1)

$b = 2a - 1 \\ </p><p>b = 2(-2) - 1 \\ </p><p>b = -4 - 1 \\ </p><p>b = (-5)$

Answer 2

Information provided with us:

• Polynomial x³ - ax² + bx - 6 leaves remainder -8
• The polynomial is divided by x - 1 and x - 2 is a factor of it.

What we have to calculate:

• Values of 'a' and 'b'

Finding out values of x,

➺ x - 1 = 0

➺ x = 1

Also,

➺ x - 2 = 0

➺ x = 2

Therefore, value of x is 2 and 1 respectively.

Putting the value of x as 1 in the given polynomial,

➺ ( 1 )³ - a( 1 )² + b( 1 ) - 6

➺ ( 1 )³ - a( 1 )² + b - 6

➺ (1 × 1 × 1) - a(1)² + b - 6

➺ ( 1 ) - a(1 × 1) + b - 6

➺ ( 1 ) - a ( 1 ) + b - 6

➺ 1 - a + b - 6

• As the remainder is -8,

➺ 1 - a + b - 6 = -8

➺ - a + b - 5 = -8

➺ - a + b = 5 - 8

➺ - a + b = -3

➺ b = a - 3

Here we got the value of b as a - 3.

Now, putting the value of x as 2 in the given polynomial:

➺ ( 2 )³ - a( 2 )² + b ( 2 ) - 6 = 0

➺ (2 × 2 × 2) - a( 2 )² + b ( 2 ) - 6 = 0

➺ (4 × 2) - a( 2 )² + b ( 2 ) - 6 = 0

➺ ( 8 ) - a( 2 )² + b ( 2 ) - 6 = 0

➺ ( 8 ) - a ( 2 × 2 ) + b ( 2 ) - 6 = 0

➺ ( 8 ) - a ( 4 ) + b ( 2 ) - 6 = 0

➺ ( 8 ) - a × 4 + b × 2 - 6 = 0

➺ 8 - 4a + 2b - 6 = 0

➺ -4a + 2b + 2 = 0

➺ -4a + 2b = -2

➺ 4a + 2b = 2

On dividing each term by 2 and multiplying each by -1 we get,

➺ 2a - b = 1

Substituting the value of b which we got,

➺ 2a - (a - 3) = 1

➺ 2a - a + 3 = 1

➺ a + 3 = 1

➺ a = 1 - 3

➺ a = -2

Now , finding out the value of b:

• Substituting the value of a in (b = a - 3)

➺ b = (-2) - (3)

➺ b = -2 - 3

➺ b = -5

Henceforth, values of a and b are -2 and -5 respectively!

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