A polynomial is a mathematical expression involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are analogous to polynomial functions in exemplar class 10 maths.
In terms of formulas, a polynomial is an expression that is equal to a linear function when graphed on the Cartesian plane.
A polynomial is an expression that can be written as a sum of linear functions, where each function in the sum has constant coefficients. In some cases, the variables “x” and “y”, maybe any real numbers or complex numbers.
A special case of a polynomial is a rational function. Such functions are defined on the set of rational numbers (or equivalently, the integers with no remainder). In this context, all polynomials over rationals are considered equivalent because they are all equal to the quotient of two polynomials over irrationals.
Most, but certainly not all, polynomials can be written as sums of linear functions. To write a polynomial as a sum of linear functions, we need to be able to create the coefficients for each function. We can find the coefficients by adding the exponents of each function in the sum.
For example, we can find the coefficient for.
The coefficient is.
The coefficients are called “coefficients” (or “constants”). The exponents are called “coefficients.” (Similarly with any degree of whatever polynomial). A more precise definition is that there are two exponents corresponding to each variable divided by the same non-zero constant: i.e.If a polynomial formula_1 has the property that, when all of its variables are written as exponents, then its degree is the sum of their exponents, then it is called a homogeneous polynomial; otherwise, it is called an inhomogeneous polynomial.
The coefficient of a given variable in a given term of the polynomial is equal to the exponent that the variable receives in that term. So, for example, in the expression formula_2 we can find:
These are important properties, and they are used often.
Coefficients can be negative. Note the time that “the exponent of the variable” is being subtracted from 1, it is a normal subtraction:
What this becomes when written as an algebraic expression is:
This can be proven by expanding the (1−x) out and subtracting 1:
The most important cases of polynomials are quadratic and linear functions. They are used over and over. Even before those, there were constants, like 2 or 5; we do not “generally” use them by themselves, but they are used to make other polynomials into a pattern that we already understand. A quadratic function would be x2+3x+2. This is a polynomial with two terms. A linear function would be: y = 2x. This has only one term, so it is a constant.
Polynomials can be manipulated in many ways, like factoring or completing the square.
Factoring”: if you have a quadratic function like:
It can be written as:
You can factor that to get:
This is done by using the distributive law. As you put numbers in for x and y, you can use that to create a pattern of constants. This process is sometimes called “factoring.” (Also, see Binomial Theorem)
“Completing the Square”:
Polynomials can also be manipulated in another way. For example, if we wanted to find the zeroes of a quadratic function, we could use the Quadratic Formula, but we don’t really need to know how to do that right now. Instead, we can manipulate this polynomial to a form where we can use the quadratic formula (or the factoring). First, look at this polynomial.
We want to make it into a form that will be easier to use. To do that, we’re going to make it into a perfect square; this is called “completing the square”. We do that by adding a term of x to both sides:
To get the other side of the equation even, we will add x2. This is called “adding equal amounts” or “adding to both sides.” To get it even, we will have to subtract 2x from both sides.
This is how we complete the square.
“General to Particular” Method:
To find the zero(es) of a general polynomial, we may be able to rewrite it as a linear (or quadratic) function, then solve for its zeroes. In other words, we go from general to particular using this method. Rewritten as a linear function, our polynomial might look like this:
where formula_5 is any real number. Solving this linear difference equation with initial conditions of formula_6 and formula_7 gives:
Hence its zeroes are given by solving (9):
In other words, the zeroes are formula_13, i.e., the roots of a quadratic equation with two imaginary constants.
Polynomials are actually very useful in nature.In biology, they are used to describe the growth of organisms over time – the size of the organism increases by a constant amount each time step.In physics, they can be used to describe any curve like an ellipse or parabola.
Also, when solving polynomial equations, there is no real way to solve them exactly… so we just approximate it and see how far off we get until we see a point that matches what we want. Infinity Learn will provide class 10 maths ncert exemplar solution.