Question

if x power4 - 2 x cube+ 3 x square - a x + b is divided by x minus 1 and X + 1 leaves remainder 5 and 19 find a and b​

Answer 1

Answer:

x⁴ -2x³ +3x² -ax +b

g(x) = x-1 and x+1

b+ a = 25

b-a = 3

= 2b = 28

b = 14

and 14 + a = 25

a = 11

Answer 2

Answer:

"Values of a and b are 5 and 8 respectively"

Step-by-step explanation:

Problem Given:

If x power4 - 2 x cube+ 3 x square - a x + b is divided by x minus 1 and X + 1 leaves remainder 5 and 19 find a and b​

Solution:

To Find:

a and b

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First, of all there is a theorem called “Polynomial Remainder Theorem”.

It is written as -  

"A Polynomial f(x) if divided by a linear polynomial (x-a) leaves remainder which equals f(a)."

Now your question,

⇒f(x) = x^4 - 2x^3 + 3x^2 - ax + b

So,When we divided by it (x - 1) it will leave remainder = f(1) = 5 (Given)

⇔f(1) = 1^4 - 2×1^3 + 3×1^2 - a×1 + b = 5 (Remainder)

⇒1 - 2 + 3 - a + b = 5

⇒a - b = (-3) ….................(Equation 1)

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Now its similarly,

f(-1) = (-1)^4 - 2×(-1)^3 + 3×(-1)^2 - a×(-1) + b = 19

⇒ 1 + 2 + 3 + a + b = 19

⇒a + b = 13 …................... (Equation 2)

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Now, Its clear we have 2 (two equations)

1.a - b = (-3)

2.a + b = 13

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According to theorem we will add these two equations,

⇔(a+b) + (a-b) = (-3) + 13

⇒ 2a = 10 => a = 5

≅So, (a +b) = 13 implies b = 8

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After add we get-

Values of a and b are 5 and 8 respectively.

Done!

Note:

In this question we use a theorem called  Polynomial Remainder Theorem” or “ Bezout’s Theorem”