10 Find the values of $a$ and $b$ and factorise the polynomial $P(x)=x^{3}+a x^{2}-x+b$, given that $P(x)$ is divisible by $x-1$ and $x+3$.
11 a Show that $x-a$ is a factor of $x^{n}-a^{n}$, for any constant $a$ and any natural number $n$.
b Find conditions (if any) on $n$ that are required in order that:
i $x+a$ is a factor of $x^{n}+a^{n}$
ii $x+a$ is a factor of $x^{n}-a^{n}$.
12 The polynomial $P(x)$ has a remainder of 2 when divided by $x-1$ and a remainder of 3 when divided by $x-2$. The remainder when $P(x)$ is divided by $(x-1)(x-2)$ is $a x+b$, i.e. $P(x)$ can be written as $P(x)=(x-1)(x-2) Q(x)+a x+b$.
a Find the values of $a$ and $b$.
b i Given that $P(x)$ is a cubic polynomial with coefficient of $x^{3}$ being 1 , and -1 is a solution of the equation $P(x)=0$, find $P(x)$.
ii Show that the equation $P(x)=0$ has no other real solutions.

Question

10 Find the values of $a$ and $b$ and factorise the polynomial $P(x)=x^{3}+a x^{2}-x+b$, given that $P(x)$ is divisible by $x-1$ and $x+3$.
11 a Show that $x-a$ is a factor of $x^{n}-a^{n}$, for any constant $a$ and any natural number $n$.
b Find conditions (if any) on $n$ that are required in order that:
i $x+a$ is a factor of $x^{n}+a^{n}$
ii $x+a$ is a factor of $x^{n}-a^{n}$.
12 The polynomial $P(x)$ has a remainder of 2 when divided by $x-1$ and a remainder of 3 when divided by $x-2$. The remainder when $P(x)$ is divided by $(x-1)(x-2)$ is $a x+b$, i.e. $P(x)$ can be written as $P(x)=(x-1)(x-2) Q(x)+a x+b$.
a Find the values of $a$ and $b$.
b i Given that $P(x)$ is a cubic polynomial with coefficient of $x^{3}$ being 1 , and -1 is a solution of the equation $P(x)=0$, find $P(x)$.
ii Show that the equation $P(x)=0$ has no other real solutions.

Accepted Answer

Hi Jack78375! It looks like you have a few questions about polynomial factorization and remainders. Let's start with the first part of your question. To find the values of a and b and factorize the polynomial P(x), we can use the given information that P(x) is divisible by (x-1) and (x+3). This means that (x-1) and (x+3) are factors of P(x). We can use polynomial long division or synthetic division to find the quotient and then factorize the polynomial. Let's work through this step by step.

Sia