Question

1. match each expression with its factored form.

i. $25m^2 - 9n^2$ a. $(5m + 3n)^2$
ii. $25m^2 - 30mn + 9n^2$ b. $(5m - 3n)^2$
iii. $25m^2 - 30mn - 9n^2$ c. $(5m + 3n)(5m - 3n)$
iv. $25m^2 + 30mn + 9n^2$ d. does not factor

Explanation:

Step1: Factor difference of squares

$25m^2 - 9n^2 = (5m)^2 - (3n)^2 = (5m+3n)(5m-3n)$

Step2: Factor perfect square trinomial

$25m^2 -30mn +9n^2 = (5m)^2 -2(5m)(3n)+(3n)^2=(5m-3n)^2$

Step3: Check factorability of trinomial

Discriminant: $(-30)^2-4(25)(-9)=900+900=1800$, not a perfect square, so it does not factor over integers.

Step4: Factor perfect square trinomial

$25m^2 +30mn +9n^2 = (5m)^2 +2(5m)(3n)+(3n)^2=(5m+3n)^2$

Answer:

I. $25m^2 - 9n^2$ → C. $(5m + 3n)(5m - 3n)$
II. $25m^2 - 30mn + 9n^2$ → B. $(5m - 3n)^2$
III. $25m^2 - 30mn - 9n^2$ → D. does not factor
IV. $25m^2 + 30mn + 9n^2$ → A. $(5m + 3n)^2$