Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Algebra, Exercise 8: END-OF-CHAPTER REVIEW EXERCISE 1

The polynomial $5x^3-13x^2+17x-7$ is denoted by $p\left(x\right)$. Find the quotient when $p\left(x\right)$ is divided by $x-1$, and show that remainder is $2$.

Show that the polynomial $5x^3-13x^2+17x-9$ has exactly one real root.

The polynomial $x^3+3x^2+4x+2$ is denoted by $f\left(x\right)$. Find the quotient and remainder when $f\left(x\right)$ is divided by $x^2+x-1$.

The polynomial $x^3+3x^2+4x+2$ is denoted by $f\left(x\right)$. Use the factor theorem to show that $x+1$ is a factor of $f\left(x\right)$.

The polynomial $4x^3+a{x}^2+9x+9$, where $a$ is constant and is denoted by $p\left(x\right)$. It is given that when $p\left(x\right)$ is divided by $2x-1$ the remainder is $10$.

Find the value of $a$ and hence verify that $(x-3)$ is a factor of $p\left(x\right)$.

The polynomial $4x^3+a{x}^2+9x+9$, where $a$ is constant and is denoted by $p\left(x\right)$. It is given that when $p\left(x\right)$ is divided by $2x-1$ the remainder is $10$ and $(x-3)$ is a factor of $p\left(x\right)$. Solve the equation $p\left(x\right)=0$

The polynomial $a{x}^3-5x^2+bx+9$, where $a$ and $b$ are constants, is denoted by $p\left(x\right)$. It is given that $2x+3$ is a factor of $p\left(x\right)$, and that when $p\left(x\right)$ is divided by $x+1$ the remainder is $8$ . Find the values of $a$ and $b$.

The polynomial $a{x}^3-5x^2+bx+9$, where $a$ and $b$ are constants, is denoted by $p\left(x\right)$. It is given that $2x+3$ is a factor of $p\left(x\right)$, and that when $p\left(x\right)$ is divided by $x+1$ the remainder is $8$. Factorise $p\left(x\right)$ completely.