Which of the binomials below is a factor of this trinomial? x2 + 4x - 32 a. x + 4 b. x - 8 c. x - 4 d. x2 + 8

The best way to do this would be to use synthetic division.  It's almost impossible to illustrate here, but maybe we can talk through it.  It's an easy process.  The first thing you have to realize is that the rules for synthetic division is that the factor you are testing HAS to be a binomial.  It cannot be something like 3x-2.  The coefficient on the x has to be a 1.  We will divide then what x is equal to into the coefficients of the polynomial.  The first one is x + 4.  If we set that equal to 0 and solve for x, we would have x = -4.  So -4 will be our first test number.  If, after we do the division, there is a 0 remainder, then x + 4 is a factor of that polynomial.  -4 is outside a "box" and inside the box are the coefficients of the polynomial.  1  4   -32.  First bring down the first number, the 1, then multiply it by the -4 to get -4.  Put that -4 up under the 4 and add to get a 0.  Now multiply that 0 by the -4 in the box and get a 0.  Put that 0 up under the -32 and add to get a remainder of -32.  That means that x + 4 is not a factor.  I'm going to jump down to C, x - 4.  If x - 4 = 0, then x = 4.  So 4 goes outside the box and the same coefficients go inside, bring down the 1 and multiply by 4 to get 4.  That 4 goes up under the 4 and add to get 8.  Multiply 8 by 4 to get 32.  Put that 32 up under the -32 and add to get a remainder of 0.  That means that x - 4 is a factor.  The numbers you get when you add, the 1 and 8 form a depressed polynomial that is 1 degree lower than the polynomial you started with.  So our first factor is x - 4, and the other is x + 8.  Very cool thing, synthetic division!!!

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