Polynomial functions, factor and remainder theorems, sum and product of roots

(a) Find the remainder when $x^3-2x^2+x-5$ is divided by $(x-2)$. [2]

(b) Given that $(x-3)$ is a factor of $x^3-4x^2+px+6$, find the value of $p$. [2]

Let $P(x)=2x^3-3x^2-11x+6$.

(a) Show that $(x-3)$ is a factor of $P(x)$. [2]

(b) Factorise $P(x)$ completely. [3]

(c) Hence solve $P(x)=0$. [1]

The polynomial $P(x)=x^3+ax^2+bx-6$ has remainder 3 when divided by $(x-1)$ and has $(x+2)$ as a factor. Find the values of $a$ and $b$.

The roots of $2x^3-7x^2+4x+3=0$ are $\alpha,\beta,\gamma$.

(a) Write down the values of $\alpha+\beta+\gamma$ and $\alpha\beta+\beta\gamma+\gamma\alpha$. [2]

(b) Find the value of $\alpha^2+\beta^2+\gamma^2$. [3]

The roots of $2x^2-5x+1=0$ are $\alpha$ and $\beta$. Find a quadratic equation with integer coefficients whose roots are $\alpha^2$ and $\beta^2$.

The diagram shows part of the graph of a cubic function $y=f(x)$. The graph touches the $x$-axis at $(-2,0)$, crosses the $x$-axis at $(3,0)$ and meets the $y$-axis at $(0,6)$.

(a) Write down the multiplicity of the zero at $x=-2$. [1]

(b) Find an expression for $f(x)$ in the form $f(x)=a(x-p)^2(x-q)$, stating the values of $a$, $p$ and $q$. [4]

When the polynomial $P(x)$ is divided by $(x-1)$ the remainder is $4$. When $P(x)$ is divided by $(x+3)$ the remainder is $-8$.

Find the remainder when $P(x)$ is divided by $(x-1)(x+3)$.

The three roots of the equation $x^3-14x^2+kx-64=0$, where $k\in\mathbb{R}$, form a geometric sequence. Let the roots be $\dfrac{\alpha}{r}$, $\alpha$ and $\alpha r$.

(a) Write down the value of the product of the roots. [1]

(b) Show that $\alpha=4$. [2]

(c) Find the value of $k$. [2]

(d) Hence find the other two roots. [2]

The polynomial $P(x)=x^3+ax^2+bx+c$ has real coefficients. One zero of $P(x)$ is $1+2\mathrm{i}$. When $P(x)$ is divided by $(x-2)$ the remainder is $-5$.

(a) Write down a second zero of $P(x)$. [1]

(b) Find the values of $a$, $b$ and $c$. [6]

Let $f(x)=x^4-3x^3+px^2+qx-8$, where $p,q\in\mathbb{R}$. When $f(x)$ is divided by $(x-1)$ the remainder is $-6$, and $(x+1)$ is a factor of $f(x)$.

(a) Find the value of $p$ and the value of $q$. [4]

(b) Use your graphic display calculator to find the real zeros of $f$. [2]

(c) Hence, find the product of the two non-real zeros of $f$. [2]

(d) Write down the set of values of $x$ for which $f(x)\le 0$. [1]