if on dividing the polynomial f(x)= 2x⁴-3x²-ax+b by x-1 and x+1 the remainders are -3 and 7 respectively then find the remainder when f(x) is divided by x-2
Answers
Answer:
22 is the remainder when f(x) is divided by x - 2
Step-by-step explanation:
According to the remainder theorem, If a polynomial p(x) is divided by g(x), then the remainder is p(a), where a is the zero of g(x)
Now, to find the remainder when f(x) is divided by x - 2, we will first need to find the value of the variables a and b.
If f(x) = 2x⁴- 3x²- ax + b is is divided by x - 1, then the remainder is 3
Zero of x - 1
⇛ x - 1 = 0
⇛ x = 0 + 1
⇛ x = 1
f(x) = 2x⁴- 3x²- ax + b = 3
f(1) = 2(1)⁴- 3(1)²- a(1) + b = 3
⇛ 2 × 1 - 3 × 1 - a + b = 3
⇛ 2 - 3 - a + b = 3
⇛ -1 - a + b = 3
⇛ -a + b = 3 + 1
⇛ -a + b = 4 ----( 1 )
If f(x) = 2x⁴- 3x²- ax + b is is divided by x + 1, then the remainder is 7
Zero of x + 1
⇛ x + 1 = 0
⇛ x = 0 - 1
⇛ x = -1
f(x) = 2x⁴- 3x²- ax + b = 7
f(-1) = 2(-1)⁴- 3(-1)²- a(-1) + b = 7
⇛ 2 × 1 - 3 × 1 + a + b = 7
⇛ 2 - 3 + a + b = 7
⇛ -1 + a + b = 7
⇛ a + b = 7 + 1
⇛ a + b = 8 ----(2)
Let's find the value of a and b
Let's find the value of a and bAdding equation (1) and (2)
⇛ (a + b) + (-a + b) = 8 + 4
⇛ a + b - a + b = 8 + 4
⇛ b + b = 8 + 4
⇛ 2b = 8 + 4
⇛ 2b = 12
⇛ b = 12/2
⇛ b = 6
Value of b is 6
⇛ a + b = 8 ( from eq 2 )
⇛ a + (6) = 8
⇛ a = 8 - 6
⇛ a = 2
Value of b and a is 6 and 2 respectively.
The remainder when f(x) is divided by x-2:
Zero of x - 2
⇛ x - 2 = 0
⇛ x = 0 + 2
⇛ x = 2
p(x) = 2x⁴- 3x²- ax + b
p(2) = 2(2)⁴ - 3(2)² - 2(2) + 6
⇛ 2 × 16 - 3 × 4 - 4 + 6
⇛ 32 - 12 + 2
⇛ 22
22 is the remainder when f(x) is divided by x - 2