Table of Contents

## What is meant by remainder theorem?

The Remainder Theorem Definition states that **when a polynomial is p ( a ) is divided by another binomial ( a – x ) then the remainder of the end result that is obtained is p ( x )**.

## What is the remainder theorem formula?

**p(x) = (x-c)·q(x) + r(x)**. The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.

## What is remainder theorem explain with example?

**an approach of Euclidean division of polynomials**. … For example: if f(a) = a

^{3}-12a

^{2}-42 is divided by (a-3) then the quotient will be a

^{2}-9a-27 and the remainder is -123.

## What is remainder theorem for Class 10?

According to the remainder theorem **if is divided by then the remainder is given by** If is divided by then the remainder is given by Hence a polynomial when divided by leaves a remainder 3 and when divided by leaves a remainder 1. Then if the polynomial is divided by it leaves a remainder .

## What is remainder theorem in Class 9?

Remainder theorem: **Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number**. If p(x) is divided by the linear polynomial x – a then the remainder is p(a). Proof: Let p(x) be any polynomial with degree greater than or equal to 1.

## What is remainder theorem formula Class 9?

**polynomial f(x) is divided by a linear polynomial [left**( x-a right)] then the remainder of that division will be equal to f(a). … 7 divided by 2 equals 3 with remainder 1 where 7 is dividend 2 is divisor 3 is quotient and 1 is remainder. [therefore 7=2times 3+1].

## How do you solve the remainder theorem?

## Why does remainder theorem work?

That is when you divide by “x – a” your remainder will just be some number. The Remainder Theorem then **points out the connection between division and multiplication**. For instance since 12 ÷ 3 = 4 then 4 × 3 = 12. If you get a remainder you do the multiplication and then add the remainder back in.

## How do you evaluate the remainder theorem?

## What is factor theorem method?

According to factor theorem if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number then **(x-a) is a factor of f(x)** if f(a)=0. … Also we can say if (x-a) is a factor of polynomial f(x) then f(a) = 0. This proves the converse of the theorem.

## What does factor theorem states?

In algebraic math the factor theorem is a theorem that **establishes a relationship between factors and zeros of a polynomial**. … Thus the factor theorem states that a polynomial has a factor if and only if: The polynomial x – M is a factor of the polynomial f(x) if and only if f (M) = 0.

## What is difference between factor theorem and remainder theorem?

The remainder theorem tells us that for any polynomial f(x) if you divide it by the binomial x−a the remainder **is equal to the value of f**(a) . The factor theorem tells us that if a is a zero of a polynomial f(x) then (x−a) is a factor of f(x) and vice-versa.

## What is factor theorem and remainder theorem Class 9?

x – a is a factor of the polynomial p(x) if p(a) = 0. Also if x – a is a factor of p(x) then p(a) = 0 where a is any real number. This is an extension to remainder theorem where remainder is 0 i.e. p(**a**) = 0.

## How do you find the remainder theorem and factor theorem?

**Remainder Theorem**

**and Factor Theorem**

- f(x) ÷ d(x) = q(x) with a remainder of r(x)
- f(x) = (x−c)·q(x) + r(x)
- f(x) = (x−c)·q(x) + r.

## Why is the factor theorem useful?

We can use the Factor Theorem **to completely factor a polynomial into the product of n factors**. Once the polynomial has been completely factored we can easily determine the zeros of the polynomial.

## How do you use the remainder theorem to find zeros?

## How can you use the remainder theorem to evaluate polynomials?

Explanation: We use the remainder theorem to establish what the remainder is **when we divide a polynomial function by a linear factor**. We can also use the remainder theorem to establish a value of f(a) . as the remainder theorem tells us that is we divide f(x) by a linear factor (x−a) the remainder is f(a) .

## Who discovered remainder theorem?

**Etienne Bezout** has discovered remainder theorem.

## What topic is factor theorem?

In algebra the factor theorem is a theorem **linking factors and zeros of a polynomial**. It is a special case of the polynomial remainder theorem.

## Which of the following statements is the remainder theorem?

The remainder theorem states the following: **If you divide a polynomial f(x) by (x – h) then the remainder is** f(h). The theorem states that our remainder equals f(h). Therefore we do not need to use long division but just need to evaluate the polynomial when x = h to find the remainder.

## What does factor theorem and Remainder Theorem mean?

## What is factor theorem in determinants?

If f(x) is a polynomial and f(α) = 0 the (x- α) is a factor of f(x). If a determinant is a polynomial in x then **(x- α)** is factor of the determinant if its value is zero when we put x = α. Using this rule we can find determinant as a product of its factors.

## What is the remainder theorem for dividing polynomials?

If a polynomial f(x) is divided by x−a the remainder is the constant f(a) **and f(x)=q(x)⋅(x−a)+f(a)** where q(x) is a polynomial with degree one less than the degree of f(x) . Synthetic division is a simpler process for dividing a polynomial by a binomial.

## Why Chinese remainder theorem is used?

**for computing with large integers**as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.

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## Is Sun Tzu a mathematician?

Sun Tzu or Sun Zi was **a Chinese mathematician of the third century CE**. His interests were in astronomy. … He is best known for authoring Sun Tzu Suan Ching (pinyin: Sun Zi Suan Jing literally “Sun Tzu’s Calculation Classic”) which contains the Chinese remainder theorem.

## What is the definition of theorem in math?

theorem in mathematics and logic **a proposition or statement that is demonstrated**. In geometry a proposition is commonly considered as a problem (a construction to be effected) or a theorem (a statement to be proved).

## What is the factor theorem A level maths?

For a polynomial f(x) the factor theorem states that: **If f(p) = 0 then (x – p) is a factor of f**(x)

## What is factor theorem in matrix?

If **each element of a matrix A is a polynomial in x and if | A | vanishes for x = a** then (x – a) is a factor of | A |. … (iii) If r rows (columns) are identical in a determinant of order n (n ≥ r) when we put x = a then (x – a)r – 1 is a factor of | A |.

## WHAT IS A if B is a singular matrix?

**if and only if its determinant is zero**. Example: Are the following matrices singular?

## What is cyclic determinant?

A cyclic operation upon the rows (or columns) of a determinant **will change the**. **value of the determinant** if the cycle is complete. Let A 8 represent the determinant formed by adding the rows of A cyclically s. in a set.

## What is the remainder?

In mathematics the remainder is **the amount “left over” after performing some computation**. In arithmetic the remainder is the integer “left over” after dividing one integer by another to produce an integer quotient (integer division).

## How do you implement Chinese remainder theorem?

**How to implement the Chinese Remainder Theorem in Java**

- What do we need to find? …
- Step 1: Find the product of all the numbers in the first array. …
- Step 2: Find the partial product of each number. …
- Find the modular multiplicative inverse of number[i] modulo partialProduct[i]. …
- Step 4: Final Sum. …
- Step 5: Return the smallest X.

## Is Chinese remainder theorem if and only if?

The Chinese remainder theorem (CRT) asserts that there is a unique class a + NZ so that x solves the system (2) if and **only if x ∈ a + NZ** i.e. x ≡ a(mod N). Thus the system (2) is equivalent to a single congruence modulo N.

## What are applications of Chinese remainder theorem in cryptography explain by giving an example?

Any k+1 people can use the Chinese remainder theorem to compute f and hence f(0) any k people do not have enough data to constrain f(0) in any way. The Chinese remainder theorem is **used to resolve multiple range ambiguities in many radar systems**.

## What is the Remainder Theorem