Question

define remainder theorem and factor theorem in polynomials​

Answer 1

Step-by-step explanation:

The factor theorem states that if f(a) = 0 for a polynomial then ( x-a) is a factor of the polynomial f(x).

If a polynomial p(x) is divided by the binomial x-a , then the remainder obtained is p(a).

Answer 2

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★ 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)