Factor f(x) = x4+ 10x3+35x2+ 50x+ 24 completely showing all work and steps with synthetic division

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It’s always a good idea to see if x=1 or -1 is a zero. The signs between the terms are all pluses so x=1 won’t be a zero, but x=-1 is: 1-10+35-50+24=60-60=0. We now know that x+1 is a factor, so we use synthetic division to reduce the quartic to a cubic:

-1 | 1 10 35  50  24

      1  -1 -9 -26 -24

      1   9 26  24 | 0

The cubic is: x³+9x²+26x+24.

We can see that x=-1 is not a zero of this cubic. So we look at the factors of 24. These include 2, 3, 4, 6. Again, all the signs are pluses, so we know we can only try negatives as zeroes. Try x=-2 first.

-8+36-52+24=-60+60=0, so x=-2 is a zero and x+2 is a factor. We now use synthetic division to reduce the cubic to a quadratic:

-2 | 1  9  26  24

      1 -2 -14 -24

      1  7  12 | 0

The quadratic is x²+7x+12, which factorises to (x+3)(x+4).

We now have all the factors: f(x)=(x+1)(x+2)(x+3)(x+4).

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