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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) ).
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 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.
If a polynomial is divided by (x - a) and the remainder equals zero then (x - a) is a factor of the polynomial.?
Yes, that's correct. According to the Factor Theorem, if a polynomial ( P(x) ) is divided by ( (x - a) ) and the remainder is zero, then ( (x - a) ) is indeed a factor of the polynomial. This means that ( P(a) = 0 ), indicating that ( a ) is a root of the polynomial. Thus, the polynomial can be expressed as ( P(x) = (x - a)Q(x) ) for some polynomial ( Q(x) ).
Which binomial is a factor of the polynomial below?
To determine which binomial is a factor of a given polynomial, you can apply the Factor Theorem. According to this theorem, if you substitute a value ( c ) into the polynomial and it equals zero, then ( (x - c) ) is a factor. Alternatively, you can perform polynomial long division or synthetic division with the given binomials to see if any of them divides the polynomial without a remainder. If you provide the specific polynomial and the binomials you're considering, I can assist further.
What is the remainder when 2 is synthetically divided into the polynomial?
To find the remainder when a polynomial is divided by (x - 2) using synthetic division, we substitute (2) into the polynomial. The remainder is the value of the polynomial evaluated at (x = 2). If you provide the specific polynomial, I can calculate the remainder for you.
What is the use of remainder and factor theorem in your daily life?
you
How do you solve a remainder theorem?
You cannot solve a theorem: you can prove the theorem or you can solve a question based on the remainder theorem.
