Question

for what the value of a would be (x+3) be a factor of x^3-4x^2+ax+27​

Answer 1

Answer:

answer is a = -12 is the correct answer

Answer 2

Answer:

$\qquad\qquad\qquad\boxed{ \sf{ \: \bf \: a = - 12 \: }}$

Step-by-step explanation:

Let assume that

$\sf \: f(x) = {x}^{3} - {4x}^{2} + ax + 27$

Further given that

$\sf \: x + 3 \: is \: a \: factor \: of \: f(x)$

So, by factor theorem, we have

$\sf \: f( - 3) = 0$

$\sf \: {( - 3)}^{3} - {4( - 3)}^{2} + a( - 3) + 27 = 0$

$\sf \: - 27 - 4(9) - 3a + 27 = 0$

$\sf \: - 36 - 3a = 0$

$\sf \: 3a = - 36$

$\sf\implies \bf \: a = - 12$

$\rule{190pt}{2pt}$

Factor theorem :-

This theorem states that if x - a is a factor of polynomial f(x) of degree greater than or equals to one, then f(a) = 0

$\rule{190pt}{2pt}$

Additional Information

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$