8x^3+84x^2+294x+343

Question

completely factor

Answer 1

343=7³ and 8=2³ so it would appear that 2x+7 could be a factor, since all the coefficients are positive. To test this, use synthetic division:

If 2x+7 is a factor, then 2x+7=0, x=-7/2 is a zero.

Try x=-7/2 as the divisor:

-7/2| 8  84  294    343

        8 -28 -196 | -343

        8  56    98 |     0 = 8x²+56x+98 with no remainder, confirming that 2x+7 is a factor.

Let's rewrite the original equation: (8x³+84x²+294x+343)/2=4x³+42x²+147x+343/2. The zeroes of this polynomial are the same those in the original polynomial. We can similarly halve the quotient: 4x²+28x+49.

2x+7 is a factor of this polynomial. So now we have to have to factorise the quadratic 4x²+28x+49=(2x+7)².

So the original polynomial is (2x+7)³.