IB Mathematics Analysis and Approaches Topic AHL…

Consider the quadratic function $f(x)=x^{2}-2 k x+k^{2}-3$ , where $k \in \mathbb{R}$ , with roots $\alpha$ and $\beta$ .

Express the sum and product of the roots of $f(x)=0$ in terms of $k$ .

Show that $α^{2} + β^{2} = 2 (^{}$ .

A polynomial $P(x)$ leaves remainders $3$ , $5$ and $11$ when divided by , and respectively.

Let $f(x)=3x^4+ax^3+bx^2+cx+d$ , where , and suppose is a factor of . Additionally, it is given that and the sum of the roots of is .

The equation $x^{3} + px^{2} + qx + r = 0$ has roots , and , and .

Consider the polynomial $P(t)=t^4-5t^3+at^2+bt+c$ , where .

Given that $\alpha^{2}+\beta^{2}=\frac{1}{\alpha^{2}+\beta^{2}}$ , find the possible values of $k$ .

For the larger positive value of $k$ found previously, the range of values of $m$ such that $f(x)=m$ has exactly two distinct real roots.

why the remainder when $P(x)$ is divided by $(x-1)(x-2)(x-3)$ has degree at most $2$ .

Find that remainder.

Hence find the remainder when $P(x)$ is divided by $(x-1)(x-2)$ , and the remainder when it is divided by $x-3$ .

Find the values of $a$ , $b$ , $c$ , and $d$ .

Factorize $f(x)$ completely into linear factors over the real numbers.

Show that the equation with roots $s-\alpha$ , $s-\beta$ and $s-\gamma$ is $y^{3} + 2py^{2} + (p^{2}+q)y + (pq - r) = 0$ .

this to $x^{3} - 6x^{2} + 11x - 6 = 0$ , and the roots of the new equation.

Find the value of $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)$ for that cubic, and the sum of the new roots.

If $(t+1)$ is a factor of $P(t)$ , an equation involving $a$ , $b$ , and $c$ .

When $(t+1)^2$ is a factor of $P(t)$ , the value of $P'(-1)$ .

Given that $(t+1)^2$ is a factor of $P(t)$ and that $a=3$ , the values of and .

IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank