2122.41 – Perfect Square?

Show that the product (10x2−17x+3)⋅(5x2+9x−2)⋅(2x2+x−6)(10x^2-17x+3)\cdot(5x^2+9x-2)\cdot(2x^2+x-6) is a perfect square, and write its square root in factored form.

Solution

(10x2−17x+3)⋅(5x2+9x−2)⋅(2x2+x−6)=((5x−1)(2x−3))((5x−1)(x+2))((2x−3)(x+2))=((5x−1)(2x−3)(x+2))2(10x^2-17x+3)\cdot(5x^2+9x-2)\cdot(2x^2+x-6)=((5x-1)(2x-3))((5x-1)(x+2))((2x-3)(x+2))\\=((5x-1)(2x-3)(x+2))^2

It's interesting to give x a value and see how it all works out. Notice in particular what happens if x = 0.