## How can we deepen our understanding of these functions and their graphs?

### Key questions

1. 1

How are the roots of polynomials related to their equations and graphs?

2. 2

What happens when one polynomial is divided by another?

3. 3

How can we rewrite polynomials and rational functions?

4. 4

Which techniques can be used to sketch rational functions?

5. 5

How can inequalities involving polynomials and rational functions be solved?

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#### Introducing...

Resource type Title
Rich example Divide it up
Food for thought Frightening function

#### Developing...

Resource type Title
Building blocks Function builder II
Package of problems What's left?
Many ways problem Can you find... asymptote edition
Many ways problem Can you find... cubic edition
Many ways problem Inequality sets
Food for thought Happy families
Food for thought How not to solve a cubic...
Go and think about it... Ab-surder!
Bigger picture General tragedy
Bigger picture Squaring the circle

#### Review questions

Title Ref
Can we factorise $6x^3+5x^2-17x-6$ completely? R6577
Can we factorise $a^3(b-c)+b^3(c-a)+c^3(a-b)$? R7991
Can we factorise $x^4 + Ax^2 + B$? R7809
Can we find the coefficients and factorise this quartic? R9330
Can we prove that $26^n$ always ends in the same two digits? R5032
Can we prove the Arithmetic and Geometric Mean Inequality? R6546
Can we show $x^4 - px^3 - 6x^2 + px +1 =0$ always has four real roots? R9189
Can we show $(x^2+1)(x+1)=(a^2+1)(a+1)$ has just one real root? R9108
Can we sketch the graph $y=x^3-x^2-x+1$? R6612
Can we sketch the graph of $y = (4x-1)/(x+5)$? R6317
Can we sketch the reciprocal of a polynomial? R6299
Can we sketch this curve with an oblique asymptote? R5854
Can we solve $\sqrt{4x+13}-\sqrt{x+1}=\sqrt{12-x}?$ R6431
Can we solve these equations for $a, b, x$ and $y$? R6871
Does $(x-1)(x-2)\times \cdots \times (x-n) = k$ have a solution? R7017
Given this equation, what's the value of $a+b+c+d$? R6219
Given three remainders, what is the function? R5248
How many pairs of positive integers $(x,y)$ satisfy this equation? R6543
How many real roots does this cubic polynomial have? R8711
How many regions are created by the graphs $y=x^3, y=x^4$ and $y=x^5$? R5685
How many solutions are there to $(x^2+1)^{10} = 2x - x^2 - 2$? R9754
If $\sqrt{x}-\sqrt{11}=\sqrt{y}$, when is $x/y$ a maximum? R9499
If dividing by $x+a$ or $x-2a$ gives the same remainder, what's $a$? R6488
If the roots of $2x^2+kx+3=0$, are $\alpha$ and $\beta$, what is $k$? R8732
If two remainders are related can we find $a$? R8468
If we divide by $x-1$, the remainder is? R6790
Two values of $x$ that differ by $5$ satisfy $x^2 - 12x + k = 0$, what is $k$? R5949
What can we say if the roots of a cubic are in arithmetic progression? R6276
What if dividing $f(x)$ by $(x-a)(x-b)$ and by $(x-a)(x-c)$ gives the same remainder? R8682
What if the remainder when we divide by $x-k$ is $k$? R8701
What if the roots of this equation are in geometric progression? R7111
What is the remainder when $p_n (x)$ is divided by $p_{n-1} (x)$? R5276
What kind of roots does $x^3 - 30x^2 + 108x - 104 = 0$ have? R9006
When are $y < x+1, y+6x < 20, x = 5y-7$ true together? R6714
When is $x^n + 1$ divisible by $x + 1$? R7108
When is $(1-x)^n (2-x)^{2n} (3-x)^{3n} (4-x)^{4n} (5-x)^{5n}$ negative? R6674
When is $n^2 x^{2n+3} - 25n x^{n+1} + 150x^7$ divisible by $x^2-1$? R7036
When is $x^2 +1$ a factor of $(3+x^4)^n-(x^2+3)^n(x^2-1)^n$? R7243
Which of these quintic equations fit this graph? R8160