Factor condition (zero remainder)

Questions where a linear expression is stated to be a factor of the polynomial, requiring the polynomial to equal zero at a specific value.

4 questions · Easy -1.0

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You are asked to find the values of $k$ which satisfy the following conditions.
(A) The graph of $y = \mathrm { f } ( x )$ goes through the origin.
(B) The graph of $y = \mathrm { f } ( x )$ intersects with the $y$ axis at ( $0 , - 2$ ).
(C) ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$.
(D) The remainder when $\mathrm { f } ( x )$ is divided by $( x + 1 )$ is 5 .
(E) The graph of $y = \mathrm { f } ( x )$ is as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
Find the solution of the equation $\mathrm { f } ( x ) = 0$ when $k = 8$. Sketch a graph of $y = \mathrm { f } ( x )$ in this case.

Use the factor theorem to prove that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ for all values of $b$.
11
The graph of $y = \mathrm { p } ( x )$ meets the $x$-axis at exactly two points.
11 (b) (i) Sketch a possible graph of $y = \mathrm { p } ( x )$ \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-20_1084_965_1619_532} 11 (b) (ii) Given $\mathrm { p } ( x )$ can be written as $$\mathrm { p } ( x ) = ( x + 2 ) \left( x ^ { 2 } + b x + 4 \right)$$ find the value of $b$. Fully justify your answer.