Question

Find the value of k, so that polynomial x3 + 3x2 – kx – 3 has one factor (x + 3)?

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{(x+3)\;is\;a\;factor\;of\;x^3 +3x^2-kx-3}$

$\underline{\textbf{To find:}}$

$\textsf{The value of 'k'}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Factor theorem:}}$

$\boxed{\textsf{(x-a) is a factor of P(x) if and only if P(a)=0}}$

$\mathsf{Let\;f(x)=x^3 +3x^2-kx-3}$

$\textsf{Since (x+3) is a factor of f(x),}$

$\textsf{we have f(-3)=0}$

$\implies\mathsf{(-3)^3 +3(-3)^2-k(-3)-3=0}$

$\implies\mathsf{-27 +27+3k-3=0}$

$\implies\mathsf{3k-3=0}$

$\implies\mathsf{3k=3}$

$\implies\boxed{\mathsf{k=1}}$

$\underline{\textbf{Answer:}}$

$\textsf{The value of k is 1}$

$\underline{\textbf{Find more:}}$

Verify whether (x + 1), (x – 2) and (x + 3) are the factors of the polynomial x3 + 2x2– 5x – 6 without actual

division. with the explanation

https://brainly.in/question/41124775  

Answer 2

\underline{\textbf{Given:}}Given:

\mathsf{(x+3)\;is\;a\;factor\;of\;x^3 +3x^2-kx-3}(x+3)isafactorofx3+3x2−kx−3

\underline{\textbf{To find:}}To find:

\textsf{The value of 'k'}The value of ’k’

\underline{\textbf{Solution:}}Solution:

\underline{\textbf{Factor theorem:}}Factor theorem:

\boxed{\textsf{(x-a) is a factor of P(x) if and only if P(a)=0}}(x-a) is a factor of P(x) if and only if P(a)=0

\mathsf{Let\;f(x)=x^3 +3x^2-kx-3}Letf(x)=x3+3x2−kx−3

\textsf{Since (x+3) is a factor of f(x),}Since (x+3) is a factor of f(x),

\textsf{we have f(-3)=0}we have f(-3)=0

\implies\mathsf{(-3)^3 +3(-3)^2-k(-3)-3=0}⟹(−3)3+3(−3)2−k(−3)−3=0

\implies\mathsf{-27 +27+3k-3=0}⟹−27+27+3k−3=0

\implies\mathsf{3k-3=0}⟹3k−3=0

\implies\mathsf{3k=3}⟹3k=3

\implies\boxed{\mathsf{k=1}}⟹k=1

\underline{\textbf{Answer:}}Answer:

\textsf{The value of k is 1}The value of k is 1

\underline{\textbf{Find more:}}Find more:

Verify whether (x + 1), (x – 2) and (x + 3) are the factors of the polynomial x3 + 2x2– 5x – 6 without actual

division. with the explanation

https://brainly.in/question/41124775