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Class 9 (Assamese)

Course: Class 9 (Assamese)>Unit 2

Lesson 3: Remainder theorem

Remainder theorem: checking factors

Learn how to determine if an expression is a factor of a polynomial by dividing the polynomial by the expression. If the remainder is zero, the expression is a factor. The video also demonstrates how to quickly calculate the remainder using the theorem.

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• At , why didn't he take the (-) sign? Is there a reason?
• He didn't take the negative sign because you take the number that when you add it to that number, you should get zero. In this case, what plus -3 = 0?
• in School i am told to say this as FACTOR THEOREM
• Factor Theorem is a special case of Remainder Theorem. Remainder Theorem states that if polynomial ƒ(x) is divided by a linear binomial of the for (x - a) then the remainder will be ƒ(a). Factor Theorem states that if ƒ(a) = 0 in this case, then the binomial (x - a) is a factor of polynomial ƒ(x).
• I have a few questions.

If its in the form (x-a), do higher powers of x work? would (x^2-a) work?

Why isn't the negative carried over into the operation? Why does Sal take the absolute value of a?

Does it then matter if I use (x-a) or (x+a)? If I take the absolute of either, it would still give me a right?
• In this case, the factor (x-3) is solved as x=3 when you add 3 to both sides to get x alone. It is possible to do this with higher powers of x, namely to the power of 2. When you solve for a factor such as (x²+a), you get x=plus or minus the square root of a. The plus or minus comes from the basic idea that any real number to the power of 2 is positive (though this way of solving DOES also work for factors with imaginary numbers). To solve a factor with this condition, you have to use synthetic division (at least, as far as I know). You can use a graphing calculator if you need to, but you set up for synthetic division and use plus or minus square root of a to divide. You will divide twice, though, once with positive square root of a and THEN with negative square root of a (using the answer from the first division). It does not matter whether you divide with the positive or negative first. Either way, you will either end up with a remainder or you won't; and this will properly prove if the factor is or is not a factor of the polynomial. I apologize if this was difficult to understand; I am far better at explaining thing visually and through solving for proof.
• Whats the differencse between remainder theorem v.s factor theorem
• Actually the factor theorem is a special case of the remainder theorem. The remainder theorem states more generally that dividing some polynomial by x-a, where a is some number, gets you a remainder of f(a). The factor theorem is more specific and says when you use the remainder theorem and the result is a remainder of 0 then that means f(a) is a root, or zero of the polynomial. Or I guess more accurately it states that if (x-a) gets a remainder of 0 then you can factor the polynomial into the answer of the division times (x-a)

• how do you get 2x^4 to 81?
• We don't.

We evaluate 2𝑥⁴ for 𝑥 = 3, which gives us
2𝑥⁴ = 2 ∙ 3⁴ = 2 ∙ (3 ∙ 3 ∙ 3 ∙ 3) = 2 ∙ 81
• What if it was like(2x-30) and you had to see if it was a factor what would you do to the 2x
• The goal is to get the form of (x-a), with the coefficient of x being 1. Divide everything by 2 in the parenthesis only,

2x-30/2 = (x-15)
Another way of thinking about it is setting is equal to 0.
2x-30=0
2x=30
x=15
And then think how we can make x equal to zero.
(x-15)

Hopefully that helps !
• is this also known as the factor theorem?
• yes we can.check if it is a factor using the remainder theorem
(1 vote)
• How come that here a division with zero is allowed? He divides p(x) with x-3 and assumes x=3.