anonymous
  • anonymous
MEDAL AND FAN Create your own factorable polynomial with a GCF. Rewrite that polynomial in two other equivalent forms. Explain how each form was created.
Mathematics
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chestercat
  • chestercat
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anonymous
  • anonymous
@Hoslos
anonymous
  • anonymous
What does GCF mean?
anonymous
  • anonymous
greatest common factor......

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anonymous
  • anonymous
Alright! From what I understand, I will choose 2 factors which are the same so that it is the GCF. What you can randomly do is choose 3 factors in the form (x-a)(bx-c)(x-a). Say, I choose (x-3)(2x-1)(x-3). Multiply them until you get the polynomial: \[(2x ^{2}-x-6x+3)(x-3)=(2x ^{2}-7x+3)(x-3)=2x ^{3}-6x ^{2}-7x ^{2}+21x+3x-9\] \[2x ^{3}-13x ^{2}+24x-9\] Now this is one way of writing the polynomial. The other way of writing it is by finding the factors of the polynomial. To do so, we have to find the factors of the last value of the polynomial. In this case is a 9 The factors are \[\pm1;\pm3;\pm9\] By trial, we plot one of the values into the polynomial to get a 0. If we try with say +3, we get \[2(2)^{3}-13(3)^{2}+24(2)-9=0\] Therefore, this is a factor. Now we know that one of the ways of writing the polynomial is (x-3). To find the other part, we have to consider the degree of the maximum variable from the main polynomial, in this case is X^3. Given that a factor is (x-3), it must multiply with a x^2 to give x^3. Plus, we have to consider the coefficient of such variable, which is 2. So we get (x-3)(2x^2+tx+c) To facilitate more, we find c, which is found by dividing the coefficient of the first factor from the last value of the original polynomial (-9). The calculation for C is then -9/-3= +3. Our factors now become (x-3)(2x^2+tx+3)
anonymous
  • anonymous
To find t, we must expand the factors, but only those that give x^2. That is because previously, we worked to find for x^3, now is x^2. The result of that will be equal to the coefficient of x^2 of the original polynomial (-13). We will get :\[-13x ^{2}=-6x ^{2} +tx ^{2}\] After cancelling out x^2, we get t=-13+6=-7 We finally have the second factor: (2x^2-7x+3) Hence , the other way of writing the polynomial is \[(x-3)(2x ^{2}-7x+3)\]

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