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\begin{align*}&\quad \, \, (10x^2-17x+3) \cdot (5x^2+9x-2) \cdot (2x^2+x-6) \\&= (5x-1)(2x-3) \cdot (5x-1)(x+2) \cdot (2x-3)(x+2) \\&=((5x-1)(2x-3)(x+2))^2\end{align*}

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