Mathematics Admissions Test (MAT) Polynomials, Graphs and Functions Syllabus

Polynomials, Graphs and Functions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Polynomials, Graphs and Functions in Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Polynomials

   3 topics

   • The Factor and Remainder Theorems
     • Testing for factors using the factor theorem
     • Finding remainders without long division
     • Polynomial division and factoring cubics and quartics
   • Roots of Polynomials
     • Relating coefficients to sums and products of roots
     • Constructing polynomials with prescribed roots
     • Repeated roots and their effect on the graph
   • The Binomial Theorem
     • Expanding (a+b) to the n for positive integer n
     • Identifying and extracting a specific term or coefficient
2. 2

   Curve Sketching

   3 topics

   • Standard Graph Shapes
     • Lines, quadratics, cubics and higher-degree polynomials
     • Reciprocal graphs and rational functions with asymptotes
     • Square-root and modulus graphs
   • Key Features of a Graph
     • Intercepts, turning points and behaviour for large positive and negative x
     • Vertical and horizontal asymptotes
     • Symmetry: odd and even functions
   • Intersections and Tangency
     • Counting intersections of two curves algebraically
     • Conditions for a line to be tangent to a curve
3. 3

   Transformations of Graphs

   3 topics

   • Translations
     • Horizontal shifts f(x-a) and vertical shifts f(x)+b
   • Stretches and Reflections
     • Scaling af(x) and f(bx)
     • Reflections in the axes
   • Composite Transformations
     • Order of applying multiple transformations
     • Effect of transformations on roots and turning points
4. 4

   Functions

   3 topics

   • Domain, Range and Notation
     • Reading and using function notation
     • Determining domain and range from a rule or graph
   • Composition and Inverses
     • Forming and evaluating composite functions
     • Finding inverse functions and the reflection in y=x
   • The Modulus Function
     • Graphs of expressions involving absolute value
     • Solving equations and inequalities containing a modulus

Polynomials, Graphs and Functions flashcards for Mathematics Admissions Test (MAT)

20 of 49 cards from the Polynomials, Graphs and Functions deck — real questions with worked answers.

1. State the Factor Theorem for a polynomial $p(x)$.

   $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$. More generally, $(bx - a)$ is a factor if and only if $p\!\left(\frac{a}{b}\right) = 0$.

2. State the Remainder Theorem.

   When a polynomial $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$. When divided by $(bx - a)$, the remainder is $p\!\left(\frac{a}{b}\right)$.

3. How can you use the Factor and Remainder Theorems to find unknown coefficients in a polynomial?

   Substitute the known root or remainder conditions to form equations. If $(x-a)$ is a factor, set $p(a)=0$; if dividing by $(x-a)$ leaves remainder $r$, set $p(a)=r$. Solve the resulting simultaneous equations for the unknowns.

4. For a quadratic $ax^{2} + bx + c = 0$ with roots $\alpha$ and $\beta$, what are the sum and product of the roots?

   $\alpha + \beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$.

5. For a cubic $ax^{3} + bx^{2} + cx + d = 0$ with roots $\alpha, \beta, \gamma$, give the three symmetric functions of the roots.

   $\alpha+\beta+\gamma = -\dfrac{b}{a}$, $\quad \alpha\beta+\beta\gamma+\gamma\alpha = \dfrac{c}{a}$, $\quad \alpha\beta\gamma = -\dfrac{d}{a}$.

6. What does the discriminant $\Delta = b^{2} - 4ac$ tell you about the roots of $ax^{2}+bx+c=0$?

   If $\Delta > 0$: two distinct real roots. If $\Delta = 0$: one repeated real root. If $\Delta < 0$: no real roots (two complex conjugate roots).

7. How is $\alpha^{2} + \beta^{2}$ expressed in terms of the sum and product of the roots?

   $\alpha^{2} + \beta^{2} = (\alpha+\beta)^{2} - 2\alpha\beta$.

8. For real-coefficient polynomials, what is true about complex (non-real) roots?

   Non-real roots occur in conjugate pairs: if $\alpha + \beta i$ is a root, so is $\alpha - \beta i$. Hence a real polynomial of odd degree always has at least one real root.

9. State the Fundamental Theorem of Algebra (as relevant to counting roots).

   A polynomial of degree $n$ (with complex coefficients) has exactly $n$ roots in $\mathbb{C}$, counted with multiplicity.

10. What is the general form of the Binomial Theorem for $(a+b)^{n}$ where $n$ is a positive integer?

    $$(a+b)^{n} = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r},$$ where $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$.

11. What is the formula for the binomial coefficient $\binom{n}{r}$?

    $\dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$. It gives the number of ways to choose $r$ items from $n$.

12. In the expansion of $(a+b)^{n}$, what is the general term (the term containing $b^{r}$)?

    The $(r+1)$th term is $\dbinom{n}{r} a^{\,n-r} b^{\,r}$.

13. How do the rows of Pascal's triangle relate to the Binomial Theorem?

    The entries in row $n$ (starting from row $0$) are the binomial coefficients $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, which are the coefficients in the expansion of $(a+b)^{n}$.

14. State the Binomial Series for $(1+x)^{n}$ when $n$ is any real number, and its condition for validity.

    $$(1+x)^{n} = 1 + nx + \frac{n(n-1)}{2!}x^{2} + \frac{n(n-1)(n-2)}{3!}x^{3} + \cdots,$$ valid for $|x| < 1$.

15. What is the shape and key feature of the graph $y = x^{2}$?

    A parabola, U-shaped, with a minimum (vertex) at the origin, symmetric about the $y$-axis ($x=0$).

16. Describe the shape of the cubic graph $y = x^{3}$.

    Increasing throughout, passing through the origin with a point of inflection there; goes from bottom-left to top-right and has rotational (odd) symmetry about the origin.

17. What does the graph of the reciprocal function $y = \dfrac{1}{x}$ look like?

    A hyperbola in two branches (one in each of the 1st and 3rd quadrants), with asymptotes at $x = 0$ (the $y$-axis) and $y = 0$ (the $x$-axis).

18. Describe the graph of the exponential function $y = a^{x}$ for $a > 1$.

    Always positive, passing through $(0,1)$, increasing, with the $x$-axis ($y=0$) as a horizontal asymptote as $x \to -\infty$.

19. Describe the graph of $y = \sqrt{x}$.

    Defined for $x \geq 0$, starting at the origin and increasing, getting flatter; it is the upper half of a sideways parabola.

20. How does the leading term of a polynomial determine its end behaviour (behaviour as $x \to \pm\infty$)?

    The leading term $a x^{n}$ dominates. Even $n$: both ends go the same way ($+\infty$ if $a>0$, $-\infty$ if $a<0$). Odd $n$: ends go opposite ways (with $a>0$, down on the left and up on the right).

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