Math Homework Help

Remainder Theorem
Let p(x) be any polynomial of degree n>0 ,and a any real number. If p(x) is divided by ( x-a), then the remainder is p(a).

The Remainder Theorem can be proved as follows.

Proof
Let us suppose when p(x) is divided by ( x-a), the quotient is q(x) and remainder is r(x).

So we have,
p(x)=(x-a) q(x)+r(x), where r(x)=0 or degree of r(x)< degree of x-a.
Since degree of ( x-a) is1, either r(x)=0 or degree of r(x)=0
So r(x) must be a constant,say r.

Thus for all values of x,
p(x)=(x-a) q(x)+r ……..(1), where r is a constant.

In particular, when x=a,
p(a)=0.q(x)+r
=0+r
=r
Hence the theorem.

Factor Theorem
Let p(x) be a polynomial of degree n>0. If p(a)=0 for a real number a, then (x-a) is a factor of
p(x). Conversely, if (x-a) is factor of p(x), then p(a)=0.

Proof

First part:
Let p(a)=0
Then by remander theorem, r=0
So equation (1) becomes
p(x)=(x-a) q(x)
==>(x-a) is a factor of p(x).

Second Part:
By remainder theorem,

p(x)=(x-a) q(x)+r
ie. p(x)=(x-a) q(x)+p(a)
Since (x-a) is a factor, p(a) must be zero.
This proves the theorem.

Example
Find the remainder when p(x)= x^2 +3x+1 is divided by x+1.
Determine whether( x-2) is a factor of p(x) or not.

Solution
x+1= x-(-1)
[Always check whether the divisor is in the form of (x-a)or not. Otherwise rewrite that in the form of(x-a)]

So, here a=-1
There fore by remainder theorem, the required remainder is
p(a)= p(-1)
=(-1)^2+3(-1)+1
=1-3+1
=-1

We know, by factor theorem,

if (x-2) is a factor of p(x), then p(2) must be zero.

Here p(2)=2^2+3(2)+1
=4+6+1
=11 which is not equal to zero.
So (x-2) is not a factor of p(x).

Let f:A–>B be a function. Let y0 be an element in B. then y0 is called a value of f provided there is some element, x0 in A, such that y0 = f(x0); that is, y0 is a value of
the function f if it corresponds, with respect to the rule of f, to some x0 in the set A = Dom(f).

Example
Find the value of the following function when x=-2

Find the Domain and Range of this function.

Yesterday we have learned what is a function.
Today let’s discuss about range and domain of a function.

Answers

Try to do more problems from your text.

Today we can discuss a topic from functions.

Definitions:

Function
Let A anb B be two non empty sets. A function “f” from a set A to a set B is a rule so that to each element x in A there corresponds exactly one element y in B, under f ,then we say that f is a functin from A to B and write
f:A -> B

y is called the image of x under f and is denoted by f(x). x and y are respectively called the independent variable and the dependent variable. We also say that y is a function of x and write
y=f(x)

Examples:

1. In the family of circles, the area A of the circle is a function of radius r of the circle.
   Here radius r is the independent variable and area A is the dependent variable
2. The speed of a chemical reaction increases 2 times with the addition of every 5 milligrams of a catalyst. Here the amount of catalyst is the independent variable and speed of the chemical reaction is the dependent variable.

There are 10 trees spaced out equally along my street with one at each end. If I run at a steady speed from one end of the street, I can reach the 5 th tree in just 5 seconds.

How long would it take me to run down the whole street at the same pase?
( No, it is not 10 seconds)

Did you get the answer?

When I reach the 5 th tree I cover 4 trees.
So,time taken to cover 4 trees is 5 seconds.
So, time taken to cover 1 tree is 5/4 seconds.
When I reach the 10 th tree I cover 9 trees.
So, time taken to cover 9 trees( and this is same as the time taken to run down the whole street) is 9 multiplies 5/4
ie 45/4
or
11.25 seconds

Addition and Subtraction

When adding or subtracting fractions, if the denominators are same, simply add or subtract the numerators and write the denominator given.

If the denominators are different, convert each fraction so that its denominator is equal to the Lowest Common Multiple of the denominators of the given fractions. This means multiplying both the numerator and denominator by the same term. For example, say the LCM is 24 and one fraction in the problem is 5/6, then to convert that fraction to denominator of 24 you must multiply numerator and denominator by4, so the fraction becomes 20/24. You follow this same procedure for each term, add or subtract as indicated by the sign of each term and divide the total by the LCM

For example:


Multiplication of Fractions is straightforward. You just multiply the numerators and multiply the denominators and then reduce the fraction to its lowest term, if possible

When we divide Fractions, we actually multiply the numerator by the reciprocal of the denominator. A reciprocal is a fraction turned upside down. For example, 3/4 divided by 5/6 = 3/4*6/5 = 18/20 = 9/10

It appears that many students, even some students of grade 9 and grade 10 are struggling with fractions. So I have prepared this lesson.Not in a thorough but simplified manner.

A term which is of the form a/b is called a fraction. The top Numbers a’ is called the numerator and the bottom number ‘b’ is called the denominator.

Proper fraction: If the numerator is less than the denominator, it is called a proper fraction
Example:1/2,4/7

Improper Fraction: If the numerator is greater than the denominator, it is called an improper fraction.

Example:5/2,9/7

Mixed Fraction: - A term consisting of an integer (5) followed by a proper fraction (2/3), written as 5 2/3 is called a mixed fraction.

see you tomorrow