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Remainder theorem: checking factors

Sal checks whether (x-3) is a factor of (2x^4-11x^3+15x^2+4x-12) using the factor theorem.

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  • old spice man green style avatar for user Maanvir Singh
    in School i am told to say this as FACTOR THEOREM
    (4 votes)
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    • mr pink red style avatar for user andrewp18
      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).
      (22 votes)
  • purple pi teal style avatar for user Kitty McGurie
    At , why didn't he take the (-) sign? Is there a reason?
    (2 votes)
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  • aqualine ultimate style avatar for user Skyler Do
    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?
    (3 votes)
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    • starky sapling style avatar for user yuzhen.henrikson.2022
      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.
      (3 votes)
  • blobby green style avatar for user SHAUN Puppala
    Whats the differencse between remainder theorem v.s factor theorem
    (3 votes)
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    • female robot grace style avatar for user loumast17
      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)

      I hope that made sense.
      (1 vote)
  • eggleston blue style avatar for user wademu
    is this also known as the factor theorem?
    (3 votes)
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  • blobby green style avatar for user Jasmine
    how do you get 2x^4 to 81?
    (2 votes)
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  • male robot donald style avatar for user Le_Go-J
    What if it was like(2x-30) and you had to see if it was a factor what would you do to the 2x
    (2 votes)
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    • leaf orange style avatar for user A/V
      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 !
      (2 votes)
  • blobby green style avatar for user drmoccy
    How come that here a division with zero is allowed? He divides p(x) with x-3 and assumes x=3.
    (2 votes)
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  • piceratops seedling style avatar for user Rain
    Would the main use of the polynomial remainder theorem be to test if something is a factor of a polynomial?
    (1 vote)
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  • piceratops ultimate style avatar for user Rose Kirby
    how do you solve a problem with two factors? like and equation where p(x) factor both (x-a) and/ or (x-b)
    (2 votes)
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    • leaf orange style avatar for user A/V
      Well there can be multiple factors in an equation/expression. However, the goal is not to find any factor, the goal is to find the *greatest common factor.* There's only one GCF so you don't have to worry about that.

      hope that helps !
      (0 votes)

Video transcript

- [Voiceover] So we're asked, Is the expression x minus three, is this a factor of this fourth degree polynomial? And you could solve this by doing algebraic long division by taking all of this business and dividing it by x minus three and figuring out if you have a remainder. If you do end up with a remainder then this is not a factor of this. But if you don't have a remainder then that means that this divides fully into this right over here without a remainder which means it is a factor. So if the remainder is equal to zero, the remainder is equal to zero, if and only if, it's a factor. It is a factor. And we know a very fast way of calculating the remainder of when you take some polynomial and you divide it by a first degree expression like this. I guess you could say when you divide it by a first degree polynomial like this. The polynomial remainder theorem, the polynomial remainder theorem tells us that if we take some polynomial, p of x and we were to divide it by some x minus a then the remainder is just going to be equal to our polynomial evaluated at our polynomial evaluated at a. So let's just see what's a in this case. Well in this case our a is positive three. So let's just evaluate our polynomial at x equals 3, if what we get is equal to zero that means our remainder is zero and that means that x minus three is a factor. If we get some other remainder that means well we have a non-zero remainder and this isn't a factor, so let's try it out. So, we're gonna have, so I'm just gonna do it all in magenta. It might be a little computationally intensive. So it's going to be two times three to the fourth power, three to the fourth, three to (mumbles), that's 81. 81. Minus 11 Yeah, this is gonna get a little computationally intensive but let's see if we can power through it. 11 times 27, I probably should have picked a simpler example, but let's just keep going. Plus 15 times nine. Plus four times three is 12. Minus 12 So lucky for us, at least those last two terms cancel out. And so this is going to be the rest from here is arithmetic. Two times 81 is 162. Now let's think about what 27 times 11 is. So let's see, 27 times 10 is going to be 270. 270 plus another 27 is minus 297. 297, did I do that, yeah, 270 So 27 times 10 is 270 plus 27, 297 Yep, that's right. And then we have, I'm prone to make careless errors here, see 90 plus 45 is 135. So plus 135. And let's see, if I were to take if I were to take 162 and 135, that's going to give me 297 minus 297. Minus 200, we do it in that green color, minus 297. And we do indeed equal zero. So the remainder, if I were to divide this by this, is equal to zero. So x minus three is indeed a factor of all of this business.