Using factor theorem, determine whether g(x) is factor of p(x) in following cases:
a) p(x) = x3 + 3x2 + 5x + 6, g(x) = x + 2
b) p(x) = 2x3 + x2
- 2x - 1, g(x) = x + 1
Answers
a)P(x) =x^3+3x^2+5x+6
g(x)=x+2=0
g(x)=x=-2
P(-2)=(-2)^3+3(-2)^2+5(-2)+6
P(-2)=-8+3(4)+5(-2)+6
P(-2)=-8+12-10+6
P(-2)=0
P(-2) is a factor of p(x) =x^3+3x^2+5x+6
b)p(x)=2x^3+x^2-2x-1
g(x) =x+1=0
g(x)=x=-1
P(-1)=2(-1)^3+(-1)^2-2(-1)-1
P(-1)=2(-1)+1+2-1
P(-1)=-2+1+2-1
P(-1)=0
P(-1) is a factor of 2x^3+x^2
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Answer :-
Using factor theorem, determine whether g(x) is factor of p(x) in following cases :
i] p(x) = x³ + 3x² + 5x + 6 , g(x) = (x + 2)
= Let's find the zero of the polynomial (x + 2),
=> x + 2 = 0
=> x = -2
Putting the value of x in p(x), we get :
=> x³ + 3x² + 5x + 6 = 0
=> (-2)³ + 3 × (-2)² + 5 × (-2) + 6 = 0
=> -8 + 3 × 4 + (-10) + 6 = 0
=> -8 + 12 - 10 + 6 = 0
=> -18 + 18 = 0
=> 0 = 0
Since, we get 0 after putting the value of x and solving the equation, g(x) is a factor of p(x).
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ii] p(x) = 2x³ + x² - 2x - 1 , g(x) = (x + 1)
= Let's the find the zero of the polynomial (x + 1),
=> x + 1 = 0
=> x = -1
Putting the value of x in p(x), we get :
=> 2 × (-1)³ + (-1)² - 2 × (-1) - 1 = 0
=> 2 × (-1) + 1 - (-2) - 1 = 0
=> -2 + 1 + 2 - 1 = 0
=> 3 - 3 = 0
=> 0 = 0
Since, we get 0 after putting the value of x and solving the equation, g(x) is a factor of p(x).