2. The Polynomial
P(x) = x4 - 2x3+3x2-ax+3a-7
When divided by (x+1), leaves the remainder 19. Find the value of a. Also, find the remainder, when p(x) is divided by (x+2)
Answers
Step-by-step explanation:
Given Polynomial ⇒
P(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7.
Divisor = x + 1
∴ x + 1 = 0
∴ x = -1
Thus,
P(-1) = (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + 3a - 7.
19 = 1 + 2 + 3 + a + 3a - 7
19 = 6 - 7 + 4a
4a - 1 = 19
4a = 20
⇒a = 5
∴ Value of a is 5.
Now, the Polynomial will be ⇒
P(x) = x⁴ - 2x³ + 3x² - (5)x + 3(5) - 7
P(x) = x⁴ - 2x³ + 3x² - 5x + 15 - 7
P(x) = x⁴ - 2x³ + 3x² - 5x + 8
Now, When this polynomial is divided by (x + 2), then,
x + 2 = 0
x = - 2
∴ P(-2) = (-2)⁴ - 2(-2)³ + 3(-2)² - 5(-2) + 8
⇒ P(-2) = 16 + 16 + 12 + 10 + 8
⇒ P(-2) = 62
Thus, Remainder will be 62.
Hope it helps.
Answer:
p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7
Divisor = x + 1
x + 1 = 0
x = -1
So, substituting the value of x = – 1 in p(x),
we get,
p(-1) = (-1)4 – 2(-1)3 + 3(-1)2 – a(-1) + 3a – 7.
19 = 1 + 2 + 3 + a + 3a – 7
19 = 6 – 7 + 4a
4a – 1 = 19
4a = 20
a = 5
Since, a = 5.
We get the polynomial,
p(x) = x4 – 2x3 + 3x2 – (5)x + 3(5) – 7
p(x) = x4 – 2x3 + 3x2 – 5x + 15 – 7
p(x) = x4 – 2x3 + 3x2 – 5x + 8
As per the question,
When the polynomial obtained is divided by (x + 2),
We get, x + 2 = 0
x = – 2
So, substituting the value of x = – 2 in p(x), we get,
p(-2) = (-2)4 – 2(-2)3 + 3(-2)2 – 5(-2) + 8
⇒ p(-2) = 16 + 16 + 12 + 10 + 8
⇒ p(-2) = 62 Therefore, the remainder = 62.
Answer:
p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7
Divisor = x + 1
x + 1 = 0
x = -1
So, substituting the value of x = – 1 in p(x),
we get,
p(-1) = (-1)4 – 2(-1)3 + 3(-1)2 – a(-1) + 3a – 7.
19 = 1 + 2 + 3 + a + 3a – 7
19 = 6 – 7 + 4a
4a – 1 = 19
4a = 20
a = 5
Since, a = 5.
We get the polynomial,
p(x) = x4 – 2x3 + 3x2 – (5)x + 3(5) – 7
p(x) = x4 – 2x3 + 3x2 – 5x + 15 – 7
p(x) = x4 – 2x3 + 3x2 – 5x + 8
As per the question,
When the polynomial obtained is divided by (x + 2),
We get, x + 2 = 0
x = – 2
So, substituting the value of x = – 2 in p(x), we get,
p(-2) = (-2)4 – 2(-2)3 + 3(-2)2 – 5(-2) + 8
⇒ p(-2) = 16 + 16 + 12 + 10 + 8
⇒ p(-2) = 62 Therefore, the remainder = 62.