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$\huge\bf{Question:-}$

Explain these topics:-

*Introduction of Polynomials
*Polynomials in One Variable
*Remainder Theorem
*Factorisation Of Polynomials
*Algebraic Identities
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Answer 1

1. Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials.

2. Polynomials in one variable are algebraic expressions that consist of terms in the form axn a x n where n is a non-negative (i.e. positive or zero) integer and a is a real number and is called the coefficient of the term. The degree of a polynomial in one variable is the largest exponent in the polynomial.

3.Remainder Theorem is an approach of Euclidean division of polynomials. According to this theorem, if we divide a polynomial P(x) by a factor ( x – a); that isn’t essentially an element of the polynomial; you will find a smaller polynomial along with a remainder. This remainder that has been obtained is actually a value of P(x) at x = a, specifically P(a). So basically, x -a is the divisor of P(x) if and only if P(a) = 0. It is applied to factorize polynomials of each degree in an elegant manner.

For example: if f(a) = a³-12a²-42 is divided by (a-3) then the quotient will be a²-9a-27 and the remainder is -123.

if we put, a-3 = 0

then a = 3

Hence, f(a) = f(3) = -123

Thus, it satisfies the remainder theorem.

4 . A polynomial can be written as the product of its factors having a degree less than or equal to the original polynomial. The process of factoring is called factorization of polynomials.

5. Algebraic Identities

Those equations of algebra which are true for every value of the variables present in the equation are as algebraic identities. The algebraic identities are also helpful in the factorization of polynomials. The utility factor in the computation of algebraic expression is found this way.

For Example: The identity (x+y)² = x²+2xy+y²will be the same for all values of x and y.

Answer 2

★ Solution :-

• Introduction of Polynomials -

This section helps us to understand the basics about Polynomials. Polynomials are the mathematical expressions which are the combination of constant terms and variable terms. These variable terms have certain degrees. On the basis of this degree, polynomials are categorised into different groups. Polynomials are basically divided into one, two and three variables.

Polynomials in different variable ::

• Polynomial in one variable = monomial

• Polynomial in two variable = binomial

• Polynomial in three variable = trinomial

• Polynomial in One Variable -

Polynomial in one variable means a polynomial expression where there is usage of only one variable. Variable terms can be many, but there is only one variable. For example : 3x + 6x + 3 = 0 . These polynomial in one variable are called as monomial.

The degree of the variable shows at how many points the graph of that expression will intersect x - axis. This degree is the highest power of tge expression.

On the basis of degree, classification of monomials ::

• One degree = Linear Polynomial

• Two degree = Quadratic Polynomial

• Three Degree = Cubic Polynomial

• Four Degree = Bi - Quadratic Polynomial

>> Euclid's Division Lemma ::

→ Dividend = Divisor × Quotient + Remainder

→ p(x) = g(x) × q(x) + r(x)

p(x) = g(x) × q(x) + r(x)→ a = bq + r

• Remainder Theorem -

According to the remainder theorem if the value of x derived from g(x), which is a polynomial which is divisor of a dividend polynomial p(x), is applied in p(x) then the result is not equal to zero.

• Factor Theorem -

According to this theorem if the valus of x derived from g(x), which is a polynomial which is divisor of a dividend polynomial p(x), is applied in p(x) then the resultant value is equal to zero.

• Factorisation of Polynomials -

Factorisation of Polynomials take place by two methods ::

>> Splitting the Middle Term - In this method, the middle term of the polynomial is split into two terms from where we start taking the common terms and finally we group the equation to get a simplest expression.

>> Factor Theorem - In this method, we firstly get the value of x from the dividend term. Then we apply this value in the expression and we equate this expression with zero which gives us the simplest form and value of variable.

• Algebraic Identities -

These are arithmetic identities which are made for convenient solving of the long expressions involving exponents. Some of the algebraic identities are given here ::

>> (a + b)² = a² + b² + 2ab

>> (a - b)² = a² + b² - 2ab

>> a² + b² = (a + b)² - 2ab

>> a² - b² = (a + b)(a - b)

>> (a + b)³ = a³ + b³ + 3ab(a + b)

>> (x + a)(x + b) = x² + (a + b)x + ab

>> a³ + b³ = (a + b)³ - 3ab(a + b)