Find K if x-1 is a factor of p(x) =x^3+8x^2-Kx-2
Answers
Answer:
Step-by-step explanation:
g(x) = x - 1 = 0
g(x) = x = 1
p(x) = x^3 + 8x^2 - Kx - 2 = 0 [given, x - 1 is a factor of p(x)]
p(1) = (1)^3 + 8(1)^2 -K(1) - 2 = 0
p(1) = 1 + 8 - K - 2 = 0
p(1) = -K + 7 = 0
p(1) = -K = -7
p(1) = K = 7
Concept:
When factoring the polynomials entirely, mathematicians apply the factor theorem. It establishes a connection between the polynomial's zeros and factors.
If f(x) is a polynomial of degree n-1 and "a" is any real number, then the factor theorem states that (x-a) is a factor of f(x), provided that f(a)=0.
We can also state that f(a) = 0 if polynomial f(x) is a factor of (x-a). This establishes the theorem's opposite.
Given:
x-1 is a factor of p(x) =x^3+8x^2-Kx-2
Find:
Find K if x-1 is a factor of p(x) =x^3+8x^2-Kx-2
Solution:
x - 1 = 0
x = 1
f(x) = x^3 + 8x^2 - Kx - 2
Using factor theorem,
f(1)= 0 [given, x - 1 is a factor of p(x)]
⇒(1)^3 + 8(1)^2 -K(1) - 2 = 0
⇒1 + 8 - K - 2 = 0
⇒-K + 7 = 0
⇒-K = -7
⇒ K = 7
Therefore, k =7
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