1. The polynomial p(x) = x+ - 2x + 3x2 - ax +3a - 7 when divided by x + 1
leaves the remainder 19. Find the values of a. Also find the remainder when
p(x) is divided by x + 2.
Answers
Answer:
Given:
Polynomial p(x)=x⁴-2x³+3x²-ax+3a-7 which gives remainder 19 when divided by x+1
To Find:
Value of 'a'.
Value of remainder when p(x) is divided by x+2
Solution:
Dividend= x⁴-2x³+3x²-ax+3a-7
Divisor= x+1
Remainder= 19
On dividing x⁴-2x³+3x²-ax+3a-7 by x+1, we get
(Calculation in First attachment)
Remainder= 4a-1
Also, it is given that
Remainder=19
⇒ 4a-1= 19
⇒ 4a= 20
⇒ a= 5
Now, after putting value of a in dividend, we get
Dividend= x⁴-2x³+3x²-(5)x+3(5)-7
Dividend= x⁴-2x³+3x²-5x+15-7
Dividend= x⁴-2x³+3x²-5x+8
Now,
Dividend= x⁴-2x³+3x²-5x+8
Divisor= x+2
After dividing x⁴-2x³+3x²-5x+8 by x+2, we get
(Calculation in second attachment)
Remainder= 62
Hence, the value of a is 5 and required remainder is 62.
Step-by-step explanation:
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.