Friends today I am going to discuss about one of the most interesting and a bit complex topic of Grade IX math that is Polynomial. Before proceeding further let’s talk about Polynomials. In earlier standards what I learn that a Polynomial is basically a term which deals in almost every type of mathematical equations or statements. The most common terminologies used in polynomial expressions are monomials, binomial and trinomials. Algebraic equations with all variables having whole number, exponents or powers are called polynomials. The expressions in which the power of variables are negative and rational numbers are not polynomials. Algebraic expression having single term known as Monomial and expression with two terms are known as Binomial whereas expressions having three terms are known as Trinomials.
Let’s see some of the examples of the polynomial equations to understand it better.
10xyz: Monomial
3x + 7y: Binomial
3x + 7y – c: Trinomial
Now considering 9th grade, I am going in deep with polynomials. In simple mathematical manner, a polynomial “p” (in any of the variable x) is an expression or a function that can assess by associating the variable and by any means by a finite number of additions, subtractions and multiplications. The most important note is that the list excludes division(for example, any number like ¾ would be considered as a constant). It also ignores radicals or values come under square root (√) (for example √5 ) considered as a constant. Most of the times we use the words or we pronounced like polynomial expression or polynomial function, but mostly we use the word Polynomial.
A polynomial can be written in the standard form as:
$displaystyle p(x) =a_nx>n + a_n-1x>n-1 + ldots + a_1 x + a_0qquad(*) $
1: P(y) = y2 – 3 is a polynomial of degree n = 2 and it is written in the standard form that is a2 = 1, a1 = 0 and a0 = -3.
2 : p(y)= by5 + 2y4 – 3y2 + 4y - √2 is a polynomial of degree 5 with a5 = b, a4 = 2, a3 = 0, a2 = -3, a1 = 4 and a0 = - √2.
Polynomial functions comes equipped with terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function, When numbers are implemented with addition or subtraction than they are said to be terms and terms are of two types Like terms or Unlike terms. Term that has the same power of the same variables is called Like term. The terms used in an expression that do not contain the same power of the same variables are called unlike terms. In an expression if the product of the numbers are used then the expression is called as factors. Variables are just representing a symbol which use different values under it whereas constant is a single value symbol.
Now I am going a bit deep with Polynomials as I am going to discuss about, how to combine polynomials. Students can add, subtract and multiply polynomials to get a new polynomial. Another point to understand is that the ratio of any two polynomials is customarily not a polynomial. Sometimes students try to memorize a number of rules for the various operations. Instead of recognition of a number of rules, student should recognize that these are just the ordinary rules underlying all algebra, such as the associative laws, distributive laws and the commutative laws of multiplication and addition. Let’s take an example to illustrate it well.
P(y) = y2 + 1 and q(y) = y – 1.
The new function t can be defined as follows (addition, subtraction and multiplication)
T = p + q where T(y) = y2 + 1 + y – 1 = y2 + y
T = p – q where T(y) = y2 + 1 – (y – 1) = y2 – y + 2
T = p x q where T(y) = (y2 + 1)(y – 1) = y3 – y2 + y - 1
One thing to remember is that students multiply powers with the same base by adding the exponents. This shows that the degree of the product of two polynomials is the sum of the individual degrees. If student adds (or subtracts) two polynomials of different degrees then the degree of the sum (or difference) is the larger of the two individual degrees. If the two polynomials have the same degrees then the degree of the sum or difference is that same degree unless the leading coefficients cancel, in which case the degree of the sum or difference is less. Student should not require to memorize these facts or rules as they need to practice a lot and make up some new examples and then think about the mechanism of the operation, and at last students need to work out the details from the understanding when student need them.
Polynomials occur all over the map in applications. Various techniques have been developed which helps us to work with polynomials and tells how to exploit their special architecture.
Now we all are going to see the basic fundamentals behind evaluating a polynomial. It can be understood more properly while taking a direct problem and try to solve the problem. Let’s take an example : p(y) = y3 – 6y2 + 11y – 6
To solve this equation if we put the value of y as 1, then the result for the polynomial is true.
P(1) = 1 – 6 + 11 – 6
P(1) = true for the polynomial.
Suppose if student wants to know what is the result for P(7). Student can evaluate P(7) in the above polynomial, multiply with the coefficients, and add the individual terms to get.
P(7) = 73 – 6 x 72 + 11 x 7 – 6
= 343 – 6 x 49 + 11 x 7 – 6
= 343 – 294 + 77 – 6
= 120.
The above procedure looks a bit clumsy. The solution or the evaluation of the above polynomial becomes much simpler if students rewrite the above polynomial p in much simpler way, as
p(y) = ((y – 6)y + 11)y – 6
Evaluating p is much easier in the above form, as students don't have to compute powers. If we do the same operation with the new form polynomial then we can get the solution faster and in better manner.
P(7) = ((7 – 6)7 + 11)7 - 6
= 18 x 7 – 6
= 120.
In the second way of solving polynomial, students get the same answer but the main difference is computation. Here the computation is simpler and it also involves some basic operations. The most important advantages of this operation or form are more pronounced or form for the polynomials of high degree. The above technique to solve polynomials work in general and it is described in the literature as synthetic division, nested multiplication or Horner's Scheme. It's a good rule of thumb that if something is known under several names it is usually powerful or otherwise important. This will be explained in another class.
Quadratic equations are also known as polynomial equations of degree two. A polynomial equation of the second order is known as Quadratic equation. The general form of quadratic equation is : ax² + bx + c = 0
where x is a variable and a, b and c are constants. Here a is quadratic coefficient, b is a linear coefficient and c is a constant term or we can say that it is a free term. Quadratic equations can be solved by using following methods: factoring, completing the squares, graphing, Newton's method, and with the help of Quadratic formula.
The Quadratic formula. Quadratic equation is ax² + bx + c = 0 and it has the solutions
here the expression under the square root sign is known as discriminant of the quadratic equation. Discriminant is denoted by the upper case Greek delta.
Delta = squared b – 4ac .
If the discriminant is zero then there is only one exact real root, also known as double root.
X = -b/2a.
The ‘±’ symbol indicate as ‘plus or minus’, which means that we need to work out the formula twice, once with a plus sign in that position, then again with a minus sign.
x=frac-b pm sqrt b>2-4ac2a,
Let’s see some of the examples of the polynomial equations to understand it better.
10xyz: Monomial
3x + 7y: Binomial
3x + 7y – c: Trinomial
Now considering 9th grade, I am going in deep with polynomials. In simple mathematical manner, a polynomial “p” (in any of the variable x) is an expression or a function that can assess by associating the variable and by any means by a finite number of additions, subtractions and multiplications. The most important note is that the list excludes division(for example, any number like ¾ would be considered as a constant). It also ignores radicals or values come under square root (√) (for example √5 ) considered as a constant. Most of the times we use the words or we pronounced like polynomial expression or polynomial function, but mostly we use the word Polynomial.
A polynomial can be written in the standard form as:
$displaystyle p(x) =a_nx>n + a_n-1x>n-1 + ldots + a_1 x + a_0qquad(*) $
Here ai are constants and also called as coefficients of the polynomial. Integer n >= 0 is called the degree of the polynomial. In most of the time this context can be stated
that
and we can adopt that convention for the purposes of this class. The form (*) shows the origin of the word polynomial which is Greek for many terms.
Any function like p(x) = 0 is also a polynomial. It is also known as zero polynomial (or we can also state it as the zero function). The degree of the function is undefined (-1 or negative infinity). Few of the examples which illustrate the above concepts: and we can adopt that convention for the purposes of this class. The form (*) shows the origin of the word polynomial which is Greek for many terms.1: P(y) = y2 – 3 is a polynomial of degree n = 2 and it is written in the standard form that is a2 = 1, a1 = 0 and a0 = -3.
2 : p(y)= by5 + 2y4 – 3y2 + 4y - √2 is a polynomial of degree 5 with a5 = b, a4 = 2, a3 = 0, a2 = -3, a1 = 4 and a0 = - √2.
Polynomial functions comes equipped with terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function, When numbers are implemented with addition or subtraction than they are said to be terms and terms are of two types Like terms or Unlike terms. Term that has the same power of the same variables is called Like term. The terms used in an expression that do not contain the same power of the same variables are called unlike terms. In an expression if the product of the numbers are used then the expression is called as factors. Variables are just representing a symbol which use different values under it whereas constant is a single value symbol.
Now I am going a bit deep with Polynomials as I am going to discuss about, how to combine polynomials. Students can add, subtract and multiply polynomials to get a new polynomial. Another point to understand is that the ratio of any two polynomials is customarily not a polynomial. Sometimes students try to memorize a number of rules for the various operations. Instead of recognition of a number of rules, student should recognize that these are just the ordinary rules underlying all algebra, such as the associative laws, distributive laws and the commutative laws of multiplication and addition. Let’s take an example to illustrate it well.
P(y) = y2 + 1 and q(y) = y – 1.
The new function t can be defined as follows (addition, subtraction and multiplication)
T = p + q where T(y) = y2 + 1 + y – 1 = y2 + y
T = p – q where T(y) = y2 + 1 – (y – 1) = y2 – y + 2
T = p x q where T(y) = (y2 + 1)(y – 1) = y3 – y2 + y - 1
One thing to remember is that students multiply powers with the same base by adding the exponents. This shows that the degree of the product of two polynomials is the sum of the individual degrees. If student adds (or subtracts) two polynomials of different degrees then the degree of the sum (or difference) is the larger of the two individual degrees. If the two polynomials have the same degrees then the degree of the sum or difference is that same degree unless the leading coefficients cancel, in which case the degree of the sum or difference is less. Student should not require to memorize these facts or rules as they need to practice a lot and make up some new examples and then think about the mechanism of the operation, and at last students need to work out the details from the understanding when student need them.
Polynomials occur all over the map in applications. Various techniques have been developed which helps us to work with polynomials and tells how to exploit their special architecture.
Now we all are going to see the basic fundamentals behind evaluating a polynomial. It can be understood more properly while taking a direct problem and try to solve the problem. Let’s take an example : p(y) = y3 – 6y2 + 11y – 6
To solve this equation if we put the value of y as 1, then the result for the polynomial is true.
P(1) = 1 – 6 + 11 – 6
P(1) = true for the polynomial.
Suppose if student wants to know what is the result for P(7). Student can evaluate P(7) in the above polynomial, multiply with the coefficients, and add the individual terms to get.
P(7) = 73 – 6 x 72 + 11 x 7 – 6
= 343 – 6 x 49 + 11 x 7 – 6
= 343 – 294 + 77 – 6
= 120.
The above procedure looks a bit clumsy. The solution or the evaluation of the above polynomial becomes much simpler if students rewrite the above polynomial p in much simpler way, as
p(y) = ((y – 6)y + 11)y – 6
Evaluating p is much easier in the above form, as students don't have to compute powers. If we do the same operation with the new form polynomial then we can get the solution faster and in better manner.
P(7) = ((7 – 6)7 + 11)7 - 6
= 18 x 7 – 6
= 120.
In the second way of solving polynomial, students get the same answer but the main difference is computation. Here the computation is simpler and it also involves some basic operations. The most important advantages of this operation or form are more pronounced or form for the polynomials of high degree. The above technique to solve polynomials work in general and it is described in the literature as synthetic division, nested multiplication or Horner's Scheme. It's a good rule of thumb that if something is known under several names it is usually powerful or otherwise important. This will be explained in another class.
Quadratic equations are also known as polynomial equations of degree two. A polynomial equation of the second order is known as Quadratic equation. The general form of quadratic equation is : ax² + bx + c = 0
where x is a variable and a, b and c are constants. Here a is quadratic coefficient, b is a linear coefficient and c is a constant term or we can say that it is a free term. Quadratic equations can be solved by using following methods: factoring, completing the squares, graphing, Newton's method, and with the help of Quadratic formula.
The Quadratic formula. Quadratic equation is ax² + bx + c = 0 and it has the solutions
here the expression under the square root sign is known as discriminant of the quadratic equation. Discriminant is denoted by the upper case Greek delta.
Delta = squared b – 4ac .
If the discriminant is zero then there is only one exact real root, also known as double root.
X = -b/2a.
The ‘±’ symbol indicate as ‘plus or minus’, which means that we need to work out the formula twice, once with a plus sign in that position, then again with a minus sign.
x=frac-b pm sqrt b>2-4ac2a,
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