Question

When f(x) = x² - 2x³ + 3x² - ax is divided by x + 1 and x - 1, we get remainders as 19 and 5 respectively. Find the remainder if f(x) is divided by x - 3.

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Answer 1

Answer:

$47$

Step By Step Explanation:

When f(x) is divided by (x + 1) and (x-1), the remainders are 19 and 5 respectively.

∴ f(-1)= 19 and f(1) = 5

⇒ (-1)⁴ -2(-1)³ + 3(-1)²-a(-1) + b = 19

⇒ 1+ 2+ 3+ a + b = 19

∴ a+b=13 -----------(i)

Again, f(1) = 5

14-2x 13+ 3 x 1²-ax1 b = 5

⇒1-2+3-a+b=5

∴b-a-3 ------------(ii)

Solving equation (i) and (ii), we get

a = 5 and b = 8

Now substituting the values of a and b in f(x), we get

∴ f(x) = x4-2x³ + 3x² - 5x +8

Now f(x) is divided by (x-3) so remainder will be f(3)

∴ f(x) = x² - 2x³ + 3x² - 5x +8[/tex]

f(3) = 34-2 x 3³+ 3x: 3x32-5x3+8

= 81-54 +27 - 15+ 8 = 47

Answer 2

Answer:

When f(x) is divided by (x + 1) and (x-1), the remainders are 19 and 5 respectively.

∴ f(-1)= 19 and f(1) = 5

⇒ (-1)⁴ -2(-1)³ + 3(-1)²-a(-1) + b = 19

⇒ 1+ 2+ 3+ a + b = 19

∴ a+b=13 -----------(i)

Again, f(1) = 5

14-2x 13+ 3 x 1²-ax1 b = 5

⇒1-2+3-a+b=5

∴b-a-3 ------------(ii)

Solving equation (i) and (ii), we get

a = 5 and b = 8

Now substituting the values of a and b in f(x), we get

∴ f(x) = x4-2x³ + 3x² - 5x +8

Now f(x) is divided by (x-3) so remainder will be f(3)

∴ f(x) = x² - 2x³ + 3x² - 5x +8[/tex]

f(3) = 34-2 x 3³+ 3x: 3x32-5x3+8

= 81-54 +27 - 15+ 8 = 47