Chapter 5 Additional Maths Adam Gibson

Identities and Equations Look at this equation: Always true An equation tells you that something is true in a particular situation, not always! Now look at this equation: Always true Certainly! The LHS and RHS are identically equal This is called an “identity”

Polynomials is not a polynomial … are mathematical expressions of the form: The above expression represents a polynomial in x of nth order, or an “nth degree polynomial”. What is a zero degree polynomial? A constant What kind of polynomial is a quadratic? A second degree polynomial Is this a polynomial in x? What order is it? For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Yes, degree 4 is not a polynomial

Some basics descending order ascending order unordered A polynomial of zero degree is a constant A polynomial of degree 1 is called linear e.g. - A polynomial of degree 2 is called quadratic e.g. – Degree 3 – cubic, 4 – quartic, 5 – quintic etc. “Degree” is also sometimes called “order” descending order ascending order unordered

Equating coefficients If: - This may be obvious, but it’s very useful! Example: Identical polynomials have identical coefficients

÷ The algebra of polynomials Polynomials can be added, subtracted or multiplied quite easily. Division is harder! Problem: Divide By ÷ Review examples of multiplication in particular, and show that it may be necessary to include an intermediate step because it is quite complicated. Students need to think about the degree of polynomial products, hopefully it will be clear with a couple of examples. The correct method for finding the solution requires thinking about two things: the degree of a product of two polynomials, and equating coefficients

2 1 To find the solution, use this identity: has degree 4 (see textbook p.9) has degree 4 is the divisor and has degree 2 The quotient, q(x), must have degree: The remainder, r(x), must have degree: 2 1 The identity can therefore be written in the following form:

What next? Equate coefficients! 3 = B 0 = C - 8A -5 = G - 8B 5 = H – 8C C=16, G=19, H=133 so, finally we get:

The “remainder” and “factor” theorems Let’s return to the formula: Now let’s consider a linear divisor: What can we say about r(x)? A: It will be a constant, so:

The remainder theorem contd. Setting x=alpha: What does this prove? It proves that if is a factor of a(x), then a(α) is zero. You can say “divides a(x) exactly”. If (x-3) divides p(x) exactly, what does it tell you about the graph of p(x)? It tells you that 3 is a root of the equation p(x)=0, or equivalently that the graph cuts the x-axis at 3.

This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after another seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life. How old was Diophantus when he died?

Diophantine equations Diophantus only looked for positive integer solutions; he considered negative and irrational solutions as foolish. the smallest solutions are: Diophantus of Alexandria - Διόφαντος ο Αλεξανδρεύς – was a Greek mathematician. He was known for his study of equations with variables which take on integer values and these Diophantine equations are named after him. Diophantus is sometimes known as the "father of Algebra". He wrote a total of thirteen books on these equations. Diophantus also wrote a treatise on polygonal numbers.

Diophantine equations Examples: Bezout’s identity It can be shown that if a and b are integers, and d is the greatest common divisor of a and b, then there exist solutions for x and y to this equation. This is a linear Diophantine equation. n=2; Pythagorean triples. n>2, no solutions. The study of Diophantine equations is part of number theory and is actually a very advanced topic.