Solve 216-756q^2+882q^4-343q^6

$$216 - 756 q ^ { 2 } + 882 q ^ { 4 } - 343 q ^ { 6 }$$

Factor

$$\left(6-7q^{2}\right)^{3}$$

Solution Steps

Find one factor of the form kq^{m}+n, where kq^{m} divides the monomial with the highest power -343q^{6} and n divides the constant factor 216. One such factor is 7q^{2}-6. Factor the polynomial by dividing it by this factor.

$$\left(7q^{2}-6\right)\left(-49q^{4}+84q^{2}-36\right)$$

Consider -49q^{4}+84q^{2}-36. Find one factor of the form pq^{u}+v, where pq^{u} divides the monomial with the highest power -49q^{4} and v divides the constant factor -36. One such factor is 7q^{2}-6. Factor the polynomial by dividing it by this factor.

$$\left(7q^{2}-6\right)\left(-7q^{2}+6\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -7q^{2}+6,7q^{2}-6.

$$\left(-7q^{2}+6\right)\left(7q^{2}-6\right)^{2}$$

Evaluate

$$\left(6-7q^{2}\right)^{3}$$