Find the values of a and b such that (x + 1) and (x – 3) are the factors of the polynomial x 3 + ax2 + 5x + b.
Answers
Step-by-step explanation:
Here is your answer :
p(x) = x³ + ax² - bx + 10
g(x) = x + 1
Put g(x) = 0
x + 1 = 0
x = - 1
g(x) is a factor of p(x).
Therefore, p(-1) = 0
{( - 1)}^{3} + a{( - 1)}^{2} - b( - 1) + 10(−1)3+a(−1)2−b(−1)+10
- 1 + a + b + 10 = 0−1+a+b+10=0
a + b + 9 = 0a+b+9=0
a = - 9 - b \: \: \: - - - - - - (1)a=−9−b−−−−−−(1)
_______________
f(x) = x + 2
Put f(x) = 0
x + 2 = 0
x = - 2
f(x) is a factor of p(x).
Therefore, p(-2) = 0
{( - 2)}^{3} + a {( - 2)}^{2} - b ( - 2) + 10 = 0(−2)3+a(−2)2−b(−2)+10=0
- 8 + 4a + 2b + 10 = 0−8+4a+2b+10=0
4a + 2b + 2 = 04a+2b+2=0
Putting the value of a from eqⁿ (1)
4( - 9 - b) + 2b + 2 = 04(−9−b)+2b+2=0
- 36 - 4b + 2b + 2 = 0−36−4b+2b+2=0
- 34 - 2b = 0−34−2b=0
- 2b = 34−2b=34
b = \frac{ - 34}{2}b=2−34
b = - 17b=−17
Put the value of b in eqⁿ (1)
a = - 9 - ( - 17)a=−9−(−17)
a = - 9 + 17a=−9+17
a = 8a=8
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Answer:
(x + 1) and (x – 3) are the factors of the polynomial x 3 + ax2 + 5x + b then
a=8
b=-17
Step-by-step explanation:
hope it helps