The polynomial p(x) = x^4-2x^3 + 3x^2-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.
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.
Answer:is given below
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
= 16 + 16 + 12 + 10 + 8
= 62
Thus, Remainder will be 62.