Polynomials Help (1 Viewer)

[Valupatitta]

Can someone please explain (in simple language PLEASE) what the following means in the syllabus and explain it

The Student is able to:

• prove that, if a polynomial has integer coefficients and if 'a' is an integer root, then 'a' is a divisor of the constant term.
• test a given polynomial with integer coefficients for possible integer roots

i've only started the topic last week and i need some1 to explain it in simple language (and most preferably with examples) on how to approach these kinds of questions that is stated in the syllabus.

tyvm ^^

 

[Zeber]

post a question.

read the textbook.

 

[Trebla]

Can someone please explain (in simple language PLEASE) what the following means in the syllabus and explain it

The Student is able to:

• prove that, if a polynomial has integer coefficients and if 'a' is an integer root, then 'a' is a divisor of the constant term.
• test a given polynomial with integer coefficients for possible integer roots

i've only started the topic last week and i need some1 to explain it in simple language (and most preferably with examples) on how to approach these kinds of questions that is stated in the syllabus.

tyvm ^^

The first dotpoint asks you to prove that a root of a polynomial can divide the constant term, if its roots are nice whole numbers. So for example (x-1)(x-2)(x-3) has a constant term of - 6, so if its roots are integers, then they are factors of 6 (e.g. roots 1,2,3 are factors of 6).

The second dotpoint asks you to basically guess integer roots using the factor theorem. So if you had an equation like x³ - 3x² + 3x - 1, then you are required to guess and check to find that say x=1 is a root of the polynomial. (i.e. when you sub it in, you get zero)

The proof for the first dotpoint is quite trivial.
Consider the general form of P(x) of degree n.
P(x) = c₁xⁿ + c₂x^n-1 + c₃x^n-2 + ............ + c_n-1x + c_n
where c_i is an integer for i = 1,2,3,.....,n
If a is an integer root of P(x) then P(a) = 0, thus
c₁aⁿ + c₂a^n-1 + c₃a^n-2 + ............ + c_n-1a + c_n = 0
=> c_n = - c₁aⁿ - c₂a^n-1 - c₃a^n-2 - ............ - c_n-1a
=> c_n = - a{c₁a^n-1 + c₂a^n-2 + c₃a^n-3 + ............ + c_n-1}
Now c_n (the constant term) and a are both integers and the big sum in the brackets is also an integer, hence we have the expression that:
c_n = a x (some integer)
This means that c_n/a = some integer, which implies that the constant term c_n is divisible by a, hence the proof is complete.

 

[independantz]

Can someone please explain (in simple language PLEASE) what the following means in the syllabus and explain it

The Student is able to:

• prove that, if a polynomial has integer coefficients and if 'a' is an integer root, then 'a' is a divisor of the constant term.
• test a given polynomial with integer coefficients for possible integer roots

i've only started the topic last week and i need some1 to explain it in simple language (and most preferably with examples) on how to approach these kinds of questions that is stated in the syllabus.

tyvm ^^

The first dot point essentially wants a proof of the integer root theorem.

proof: From what i can remember...
Let P(x)= Ax^2+Bx+C, where A,B,C are integers
P(a)=0, where a is an integer root.

i.e. A(a)^2+B(a)+C=0
therefore, -C/a=A(a)+B
-C/a=S, where S is an integer, since A,a and B are integers.
Therfore, since an integer root will divide the constant term, since C is the constant term and the integer root,a divided it to produce an integer.

The second dot point basically requires to use this theorem in a range of applications.

For example if they give you a polynomial which has an integer root, then you know that the root will be a factor of the constant term, bear in mind that it is an integer. So you sub factors of the constant term into the polynomial until you find the root...