Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Algebra, Exercise 7: EXERCISE 1F

The polynomial $f\left(x\right)=x^3+6x^2+px-3$, leaves a remainder of $R$ when divided by $x+1$ and a remainder of $-10R$ when divided by $x-3$. Find the value of $p$.

The polynomial $f\left(x\right)=x^3+6x^2+px-3$, leaves a remainder of $R$ when divided by $x+1$ and a remainder of $-10R$ when divided by $x-3$. Find the remainder when the expression is divided by $x-2$.

The polynomial $f\left(x\right)=x^3+a{x}^2+bx+2$, where $a$ and $b$ are constants. It is given that $x-2$ is a factor of $f\left(x\right)$, and that when $f\left(x\right)$ is divided by $x+1$ the remainder is $21$ . Find the value of $a$ and the value of $b$.

The polynomial $f\left(x\right)=x^3+a{x}^2+bx+2$, where $a$ and $b$ are constants. It is given that $x-2$ is a factor of $f\left(x\right)$, and that when $f\left(x\right)$ is divided by $x+1$ the remainder is $21$. Solve the equation $f\left(x\right)=0$ , giving the roots in exact form.

The polynomial $f\left(x\right)=2x^3+a{x}^2+bx+c$. The roots of $f\left(x\right)=0$ are $-1,2$ and $k$. When $f\left(x\right)$ is divided by $x-1$, the remainder is $6$. Find the value of $k$.[Write your answer as decimal only]

The polynomial $f\left(x\right)=2x^3+a{x}^2+bx+c$. The roots of $f\left(x\right)=0$ are $-1,2$ and $k$. When $f\left(x\right)$ is divided by $x-1$, the remainder is $6$. Find the remainder when $f\left(x\right)$ is divided by $x+2$.

The polynomial $P\left(x\right)=2x+4x^2+6x^3+\dots +100x^{50}$. Find the remainder when $P\left(x\right)$ is divided by $x-1$.

$P\left(x\right)=3(x+1)(x+2)(x+3)+a(x+1)(x+2)+b(x+1)+c$. It is given that when $P\left(x\right)$ is divided by each of $x+1,x+2\text{ and }x+3$ the remainders are $-2,2 \text{and} 10$, respectively. Find the value of $a$, the value of $b$ and the value of $c$.