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The polynomial $3{x^3} + 2{x^2} + \alpha x + b$, where $\alpha $ and $b$ are constants, is denoted by $p\left( x \right)$. It is given that $\left( {x - 1} \right)$ is a factor of $p\left( x \right)$, and that when $p\left( x \right)$ is divided by $\left( {x - 2} \right)$ the remainder is 10.

a) Find the values of $\alpha $ and $b$.

b) When $\alpha $ and $b$ have these values, solve the equation $p\left( x \right) = 0$.

پاسخ تشریحی :
نمایش پاسخ

a) Substitute $x = 1$, equate to zero and obtain a correct equation in any form

Substitute $x = 2$ and equate to 10

Obtain a correct equation in any form

Solve a relevant pair of equations for $\alpha $ or for $b$

Obtain $\alpha  =  - 17$ and $b = 12$

b) At any stage, state that $x = 1$ is a solution

EITHER: Attempt division by $x - 1$ and reach a partial quotient of $3{x^2} + 5x$

Obtain solutions $x =  - 3$ and $x = \frac{4}{3}$

OR: Obtain solution $x =  - 3$ by trial and error or inspection

Obtain solution $x = \frac{4}{3}$

[If an attempt at the quadratic factor is made by inspection, the M1 is earned if it reaches an unknown factor of $3{x^2} + 5x + \lambda $ and an equation in $\lambda $]

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