What Is Remainder Theorem


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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?

The remainder theorem formula is: 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?

Remainder Theorem is an approach of Euclidean division of polynomials. … 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.

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?

The remainder theorem states that when a 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].

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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
  1. f(x) ÷ d(x) = q(x) with a remainder of r(x)
  2. f(x) = (x−c)·q(x) + r(x)
  3. 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?

We get a remainder of 0 which verifies that indeed p(1)=0. Our quotient polynomial is a second degree polynomial with coefficients 2 2 and −3. So q(x)=2×2+2x−3. Theorem 3.4 tells us p(x)=(x−1)(2×2+2x−3).

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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?

The Chinese remainder theorem is widely 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?

2.5.3 Factor Theorem

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

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What is factor theorem in matrix?

Theorem 7.3 (Factor Theorem)

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?

A singular matrix is one which is non-invertible i.e. there is no multiplicative inverse B such that the original matrix A × B = I (Identity matrix) A matrix is singular 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
  1. What do we need to find? …
  2. Step 1: Find the product of all the numbers in the first array. …
  3. Step 2: Find the partial product of each number. …
  4. Find the modular multiplicative inverse of number[i] modulo partialProduct[i]. …
  5. Step 4: Final Sum. …
  6. 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

Remainder Theorem and Synthetic Division of Polynomials

Polynomial remainder theorem | Polynomial and rational functions | Algebra II | Khan Academy

Remainder Theorem class 9th

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