Answer :
To find the values of a and b in the given polynomial function with specific remainders, set up equations using the remainder theorem and solve for a and b, resulting in a = -7 and b = -41.
Given:
- f(x) = ax³ - x² - 5x + b
- Remainder when f(x) is divided by (x-2) is 36
- Remainder when f(x) is divided by (x+2) is 40
To find a and b:
- Set up equations using the remainder theorem.
- and equate both the remainder with the a and b according to given in questions.
- After solving, we get a = -7 and b = -41
Answer:
a = 1 , b = 42
Step-by-step explanation:
Using the Remainder theorem
If a polynomial f(x) is divided by (x ± a ) then the remainder is f(∓ a )
given
f(x) divided by (x - 2) with remainder 36 , and divided by (x + 2) with remainder 40, then
f(2) = 36 and f(- 2) = 40
f(x) = ax³ - x² - 5x + b
f(2) = a(2)³ - 2² - 5(2) + b = 36 , that is
8a - 4 - 10 + b = 36
8a + b - 14 = 36 ( add 14 to both sides )
8a + b = 50 → (1)
and
f(- 2) = a(- 2)³ - (- 2)² - 5(- 2) + b = 40 , that is
- 8a - 4 + 10 + b = 40
- 8a + b + 6 = 40 ( subtract 6 from both sides )
- 8a + b = 34 → (2)
Solve the 2 equations simultaneously for a and b
8a + b = 50 → (1)
- 8a + b = 34 → (2)
add (1) and (2) term by term to eliminate a
(8a - 8a ) + (b + b ) = 50 + 34
0 + 2b = 84
2b = 84 ( divide both sides by 2 )
b = 42
substitute b = 42 into either of the 2 equations and solve for a
substituting into (1)
8a + 42 = 50 ( subtract 42 from both sides )
8a = 8 ( divide both sides by 8 )
a = 1
The values of a and b are a = 1 , b = 42
