Question

Could someone please help with these three problems? They are simplifying expressions by factoring. Thank you so much!

Answer 1

6)

$\bf \cfrac{2x^2-13x+15}{x^2-2x}\cdot \cfrac{x^2-4x+4}{10-7x+x^2}\implies \cfrac{(2x-3)(x-5)}{x\underline{(x-2)}}\cdot \cfrac{(x-2)\underline{(x-2)}}{x^2-7x+10} \\\\\\ \cfrac{(2x-3)\underline{(x-5)}}{x}\cdot \cfrac{\underline{(x-2)}}{\underline{(x-2)}~\underline{(x-5)}}\implies \cfrac{(2x-3)}{x}\cdot \cfrac{1}{1}\implies \cfrac{2x-3}{x}$

7)

$\bf \textit{difference of squares} \\\\ (a-b)(a+b) = a^2-b^2\qquad \qquad a^2-b^2 = (a-b)(a+b)\\\\ -------------------------------$

$\bf \cfrac{4x^2-9y^2}{6x^2-9xy}\cdot \cfrac{6y^2}{4xy+6y^2}\implies \cfrac{2^2x^2-3^2y^2}{3x(2x-3y)}\cdot \cfrac{6y^2}{2y(2x+3y)} \\\\\\ \cfrac{(2x)^2~-~(3y)^2}{\underline{3} x(2x-3y)}\cdot \cfrac{\underline{2}\cdot \underline{3} y\underline{y}}{\underline{2y}(2x+3y)}\implies \cfrac{\underline{(2x-3y)}~\underline{(2x+3y)}}{x\underline{(2x-3y)}}\cdot \cfrac{y}{\underline{2x+3y}} \\\\\\ \cfrac{1}{x}\cdot \cfrac{y}{1}\implies \cfrac{y}{x}$

11)

$\bf \cfrac{2x-3}{5x+1}\div\cfrac{6x^2-13x+6}{15x^2-7x-2}\implies \cfrac{2x-3}{5x+1}\div\cfrac{(2x-3)\underline{(3x-2)}}{\underline{(3x-2)}~(5x+1)} \\\\\\ \cfrac{2x-3}{5x+1}\div\cfrac{2x-3}{5x+1}\implies \cfrac{\underline{2x-3}}{\underline{5x+1}}\cdot\cfrac{\underline{5x+1}}{\underline{2x-3}}\implies \cfrac{1}{1}\cdot \cfrac{1}{1}\implies 1$