Consider the polynomial
P(x)=3x⁴-ax³+2ax²-x-b
If the remainder when P(x) is divided by x+1 is 6 and 1 is a zero of P(x), what are a and b?
Answers
Answer :
The required values of a and b are 2 and 4 respectively.
Step-by-step explanation :
Given :
- P(x) = 3x⁴ - ax³ + 2ax² - x - b
- the remainder when P(x) is divided by x + 1 is 6
- 1 is a zero of P(x)
To find :
the values of a and b
Solution :
Given polynomial, P(x) = 3x⁴ - ax³ + 2ax² - x - b
⇢ The remainder theorem states when a polynomial p(x) is divided by (x - q), then the remainder is p(q)
P(x) is divided by (x + 1). then remainder is 6
Here, q = -1,
p(-1) = 6
3(-1)⁴ - a(-1)³ + 2a(-1)² - (-1) - b = 6
3(1) - a(-1) + 2a(1) + 1 - b = 6
3 + a + 2a + 1 - b = 6
4 + 3a - b = 6
3a - b = 6 - 4
3a - b = 2 ➛ [1]
⇢ Since 1 is a zero of the given polynomial, p(1) = 0
p(1) = 0
3(1)⁴ - a(1)³ + 2a(1)² - (1) - b = 0
3(1) - a(1) + 2a(1) - 1 - b = 0
3 - a + 2a - 1 - b = 0
2 + a - b = 0
a - b = 0 - 2
a - b = -2 ➛ [2]
Subtract equation [2] from equation [1],
3a - b - (a - b) = 2 - (-2)
3a - b - a + b = 2 + 2
2a = 4
a = 4/2
a = 2
Substitute a = 2, in equation [2]
a - b = -2
2 - b = -2
b = 2 + 2
b = 4
Therefore, a = 2 and b = 4
Answer:
Answer :
The required values of a and b are 2 and 4 respectively.
Step-by-step explanation :
Given :
P(x) = 3x⁴ - ax³ + 2ax² - x - b
the remainder when P(x) is divided by x + 1 is 6
1 is a zero of P(x)
To find :
the values of a and b
Solution :
Given polynomial, P(x) = 3x⁴ - ax³ + 2ax² - x - b
⇢ The remainder theorem states when a polynomial p(x) is divided by (x - q), then the remainder is p(q)
P(x) is divided by (x + 1). then remainder is 6
Here, q = -1,
p(-1) = 6
3(-1)⁴ - a(-1)³ + 2a(-1)² - (-1) - b = 6
3(1) - a(-1) + 2a(1) + 1 - b = 6
3 + a + 2a + 1 - b = 6
4 + 3a - b = 6
3a - b = 6 - 4
3a - b = 2 ➛ [1]
⇢ Since 1 is a zero of the given polynomial, p(1) = 0
p(1) = 0
3(1)⁴ - a(1)³ + 2a(1)² - (1) - b = 0
3(1) - a(1) + 2a(1) - 1 - b = 0
3 - a + 2a - 1 - b = 0
2 + a - b = 0
a - b = 0 - 2
a - b = -2 ➛ [2]
Subtract equation [2] from equation [1],
3a - b - (a - b) = 2 - (-2)
3a - b - a + b = 2 + 2
2a = 4
a = 4/2
a = 2
Substitute a = 2, in equation [2]
a - b = -2
2 - b = -2
b = 2 + 2
b = 4
Therefore, a = 2 and b = 4