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If a polynomial is divided by x - c, we can use the Remainder theorem to evaluate the polynomial at c.
The Remainder theorem:
If the polynomial f(x) is divided by x - c, then the remainder is f(c).
Example:
Given f(x) = x^3 - 4x^2 + 5x + 3, use the remainder theorem to find f(2).
Solution:
By the remainder theorem, if f(x) is divided by x - 2, then the remainder is f(2).
We can use the synthetic division to divide.
2] 1 -4 5 3
2 -4 2
__________
1 -2 1 5
The remainder is 5, so f(2) = 5
Check:
f(x) = x^3 - 4x^2 + 5x + 3
f(2) = (2)^3 - 4(2)^2 + 5(2) + 3 = 8 - 16 + 10 + 3 = 5
What else can I help you with?
What is polynomial division?
That means that you divide one polynomial by another polynomial. Basically, if you have polynomials "A" and "B", you look for a polynomial "C" and a remainder "R", such that: B x C + R = A ... such that the remainder has a lower degree than polynomial "B", the polynomial by which you are dividing. For example, if you divide by a polynomial of degree 3, the remainder must be of degree 2 or less.
If a polynomial is divided by (x-a) and the remainder equals zero then (x-a) is a factor of the polynomial?
false - apex
What is the polynomial factor theorem?
Suppose p(x) is a polynomial in x. Then p(a) = 0 if and only if (x-a) is a factor of p(x).
Is it false if a polynomial is divided by (x-a) and the remainder equals zero then (x-a) is a factor of the polynomial?
No, it’s true. It’s the same as saying if 60 is divided by 2 and the remainder equals zero (no remainder, so it divides perfectly), 2 is a factor of 60.
What is the remainder when 3 is synthetically divided into the polynomial -2x2 7x-9?
-6
Who discovered the polynomial remainder theorem?
Euclid
What is the 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.The factor theorem states that a polynomial has a factor if and only if
What does factor theorem mean?
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.The factor theorem states that a polynomial has a factor if and only if
How can you tell if a binomial divides evenly into a polynomial?
Do the division, and see if there is a remainder.
What is the remainder R when the polynomial p(x) is divided by (x - 2)?
The remainder ( R ) when a polynomial ( p(x) ) is divided by ( (x - 2) ) can be found using the Remainder Theorem. According to this theorem, the remainder is equal to ( p(2) ). Thus, to find ( R ), simply evaluate the polynomial at ( x = 2 ): ( R = p(2) ).
When the polynomial in P(x) is divided by (x plus a) the remainder equals P(a)?
When a polynomial ( P(x) ) is divided by ( (x + a) ), the remainder can be found using the Remainder Theorem. This theorem states that the remainder of the division of ( P(x) ) by ( (x - r) ) is equal to ( P(r) ). Therefore, when dividing by ( (x + a) ), which is equivalent to ( (x - (-a)) ), the remainder is ( P(-a) ), confirming that ( P(-a) ) is the value of the polynomial evaluated at ( -a ).
According to the Remainder theorem the remainder of the problem in which a polynomial F x is divided by the binomial x - a equals?
F(a)
What is the reminder theorem?
The Remainder Theorem states that for a polynomial ( f(x) ), if you divide it by a linear factor of the form ( x - c ), the remainder of this division is equal to ( f(c) ). This means that by evaluating the polynomial at ( c ), you can quickly determine the remainder without performing long division. This theorem is useful for factoring polynomials and analyzing their roots.
Can x-5 be a remainder on division of a polynomial px by 7x 2 justify your answer?
Yes, ( x - 5 ) can be a remainder when dividing a polynomial ( p(x) ) by ( 7x^2 ). According to the polynomial remainder theorem, the remainder of a polynomial division by a polynomial of degree ( n ) will have a degree less than ( n ). Since ( 7x^2 ) is a polynomial of degree 2, the remainder can be of degree 1 or less, which means it can indeed be of the form ( x - 5 ).
How can you find the remainder by using the remainder theorem?
The Remainder Theorem states that if you divide a polynomial ( f(x) ) by a linear divisor of the form ( x - c ), the remainder is simply ( f(c) ). To find the remainder, substitute the value ( c ) into the polynomial ( f(x) ) and calculate the result. The output will be the remainder of the division. This method significantly simplifies finding remainders without performing long division.
What is the remainder thereom?
The remainder theorem states that if you divide a polynomial function by one of it's linier factors it's degree will be decreased by one. This theorem is often used to find the imaginary zeros of polynomial functions by reducing them to quadratics at which point they can be solved by using the quadratic formula.
What is the remainder when you divide 4x3-5x2 3x-1 by x-2?
To find the remainder when dividing the polynomial (4x^3 - 5x^2 + 3x - 1) by (x - 2), we can use the Remainder Theorem. According to this theorem, the remainder is equal to the value of the polynomial evaluated at (x = 2). Calculating (4(2)^3 - 5(2)^2 + 3(2) - 1) gives: [ 4(8) - 5(4) + 6 - 1 = 32 - 20 + 6 - 1 = 17. ] Thus, the remainder is (17).
