What are the Basics of Polynomials in Mathematics?

According to the new CBSE Exam Pattern, Basics of Polynomials carries 20 marks.

Polynomials proves to be an important part of the “language” of mathematics and algebra. Regarding its use, they are found to be used in nearly every field of mathematics in order to express numbers as a result of mathematical operations. It should be noted that polynomials also happen to be “building blocks” in other types of mathematical expressions, like rational expressions.

In everyday life, many mathematical processes are done which can be interpreted as polynomials. When it comes to summing the cost of items on a grocery bill, it can be interpreted as a polynomial. When we calculate the distance traveled of a vehicle or object, it can be interpreted as a polynomial. Similarly calculating perimeter, area, and volume of geometric figures can be interpreted as polynomials. These are just some of the many applications of polynomials.

Polynomials can be used to plot complex curves that decides the path of missile trajectories or a roller coaster or model a complex situation in physics experiment. Polynomial modeling functions are also found to be used to solve questions in chemistry and biology.

Polynomials also proves to be an essential tool to describe and predict the traffic patterns so appropriate traffic control measures, such as traffic lights, can be implemented. Polynomials are used by Economists to model economic growth patterns, and medical researchers use them to describe the behavior of bacterial colonies.

“Polynomial” has been derived from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)- therefore, it means many terms.

A quadratic polynomial in x with real coefficients is of the form ax² + bx + c, where a, b, c proves to be the real numbers with a ≠ 0.

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 known as a linear polynomial, degree 2 is called quadratic polynomial, and degree 3 is called a cubic polynomial.

If p(x) proves to be a polynomial in x, and if k is any real number, then the value which is obtained by replacing x by k in p(x), is known as 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 earlier classes. Check out How to factor polynomials for details

This chapter we will focusing on finding the relationship between the coefficients and Zeroes of the polynomial expression. We will also study the division algorithm for polynomial

Degree – The degree of polynomial is the highest exponent of the variable in the polynomial. Example: 3×3 + 4, here degree = 3. Note that Polynomials of degrees 1, 2 and 3 are known as linear, quadratic and cubic polynomial respectively.

There can be terms in a polynomial which have Constants like 3, -20, etc., Variables like x and y and Exponents like 2 in y².

These can be combined with the use of addition, subtraction and multiplication but NOT DIVISION.

The zeroes of a polynomial p(x) proves to be precisely the x-coordinates of the points. Here the graph of y = p(x) intersects the x-axis.

If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then

Sum of zeros: α+β=−ba=−(coefficient of x) (coefficient of x2)

Product of zeros: αβ=ca=constant term coefficient of x2

If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d = 0, then

α +β+ γ=−ba=−(coefficient of x2) (coefficient of x3)

αβ +βγ+γα=ca= (coefficient of x) (coefficient of x3)

αβγ=−da=−constant term coefficient of x3

Zeroes (α, β, γ) follow the rules of algebraic identities, i.e.,

(α + β) ² = α² + β² + 2αβ

∴ (α² + β²) = (α + β) ² – 2αβ

Division algorithm:

If p(x) and g(x) proves to be any two polynomials with g(x) ≠ 0, then we get

p(x) = g(x) × q(x) + r(x)

Here, Dividend = Divisor x Quotient + Remainder

You need to note this!

If r (x) = 0, then g (x) proves to be a factor of p (x).

If r (x) ≠ 0, then r (x) can be subtracted from p (x) and then the new polynomial formed is found to be a factor of g (x) and q(x).