We have studied polynomial expression in one variable and their degrees in the previous classes.
Now we know that the highest power of x in p(x) is called the degree of the polynomial p(x).A polynomial of degree 1 is called a linear polynomial , degree 2 is called quadratic polynomial,degree 3 is called a cubic polynomial.
If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).A real number k is said to be a zero of a polynomial p(x), if p(k) = 0
Factoring of Polynomials can be done using grouping,split mid-term method,identity method. This we also learned in earliar classes.Check out How to factor polynomials for details
This chapter we will focussing on finding the relationship between the coefficients and Zeroes of the polynomials expression. We will also study the division algorithm for polynomial
Geometric Meaning of the Zero's of the polynomials
Lets us assume
y= p(x) where p(x) is the polynomial of any form.
Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained by putting the values. The plot or graph obtained can be of any shapes
The zero's of the polynomial are the points where the graph meet x axis in the Cartesian plane. If the graph does not meet x axis ,then the polynomial does not have any zero's.
Let us take some useful polynomial and shapes obtained on the Cartesian plane
N degree Polynomial
More facts about the geometric shape of the Polynomials
If the degree n of a polynomial is even, then the arms of the graph are either both up or both down.
If the degree n is odd, then one arm of the graph is up and one is down.
If the leading coefficient an is positive, the right arm of the graph is up.
If the leading coefficient an is negative, the right arm of the graph is down
These above points can also be applied to all the example polynomial given up.
These points will help in roughly drawing the graph of any polynomial
Relation between coefficient and zero's of the Polynomial:
Formation of polynomial when the zeros are given
Division algorithm for Polynomial
Let's p(x) and q(x) are any two polynomials with q(x) ≠ 0 ,then we can find polynomial s(x) and r(x) such that
P(x)=s(x) q(x) + r(x)
Where r(x) can be zero or degree of r(x) < degree of g(x)
Steps to divide a polynomial by another polynomial
Arrange the term in decreasing order in both the polynomial
Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term,
Similar steps are followed till we get the reminder whose degree is less than of divisor
Divide P(x) by q(x)
$P(x)=x^4 +x +1$
Following the step outlined above,here is the division
$x^4 +x +1=(x+1)(x^3-x^2+x)+1$
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