34.4
![\frac{16}{49} p^2q - q^3 = q* ( (\frac{4}{7}p)^2 - q^2) = q( \frac{4}{7}p -q)( \frac{4}{7} p +q) \\ \\ \\ 2 \frac{7}{9} a^3b - \frac{ab^3}{4} = \frac{25}{9} a^3b - \frac{1}{4} ab^3= ab(( \frac{5}{3}a)^2 - (\frac{1}{2} b)^2 ) = \\ \\ =ab(1 \frac{2}{3} a- \frac{1}{2} b)(1 \frac{2}{3} a+\frac{1}{2} b) =ab(1 \frac{2}{3} a- 0.5b)(1 \frac{2}{3} a+0.5b) \\ \\ \\](https://tex.z-dn.net/?f=+%5Cfrac%7B16%7D%7B49%7D+p%5E2q++-+q%5E3+%3D+q%2A+%28+%28%5Cfrac%7B4%7D%7B7%7Dp%29%5E2++-+q%5E2%29+%3D+q%28+%5Cfrac%7B4%7D%7B7%7Dp+-q%29%28+%5Cfrac%7B4%7D%7B7%7D+p+%2Bq%29+%5C%5C++%5C%5C++%5C%5C+%0A2+%5Cfrac%7B7%7D%7B9%7D+a%5E3b+-++%5Cfrac%7Bab%5E3%7D%7B4%7D+%3D++%5Cfrac%7B25%7D%7B9%7D+a%5E3b+-++%5Cfrac%7B1%7D%7B4%7D+ab%5E3%3D+ab%28%28+%5Cfrac%7B5%7D%7B3%7Da%29%5E2++-++%28%5Cfrac%7B1%7D%7B2%7D+b%29%5E2+%29+%3D+%5C%5C++%5C%5C+%0A%3Dab%281+%5Cfrac%7B2%7D%7B3%7D+a-+%5Cfrac%7B1%7D%7B2%7D+b%29%281+%5Cfrac%7B2%7D%7B3%7D+a%2B%5Cfrac%7B1%7D%7B2%7D+b%29+%3Dab%281+%5Cfrac%7B2%7D%7B3%7D+a-+0.5b%29%281+%5Cfrac%7B2%7D%7B3%7D+a%2B0.5b%29++%5C%5C++%5C%5C++%5C%5C+)
![c^3 - \frac{25}{36} cd^2 = c (c^2 - ( \frac{5}{6} d)^2 ) = c(c - \frac{5}{6} d)(c^2 + \frac{5}{6} d) \\ \\ \\ \frac{mn^5}{9} - 3 \frac{1}{16} m^3n = \frac{1}{9} *mn^5 - \frac{49}{9} m^3n = mn( \frac{1}{9} n^4 - \frac{49}{9} m^2) = \\ \\ =mn( ( \frac{1}{3} n^2)^2 - ( \frac{7}{3} m)^2 ) = mn( \frac{1}{3}n^2 - 2 \frac{1}{3}m)( \frac{1}{3}n^2 +2 \frac{1}{3}m)](https://tex.z-dn.net/?f=c%5E3++-++%5Cfrac%7B25%7D%7B36%7D+cd%5E2+%3D+c+%28c%5E2++-+%28+%5Cfrac%7B5%7D%7B6%7D+d%29%5E2+%29+%3D+c%28c++-+%5Cfrac%7B5%7D%7B6%7D+d%29%28c%5E2++%2B+%5Cfrac%7B5%7D%7B6%7D+d%29+%5C%5C++%5C%5C++%5C%5C+%0A+%5Cfrac%7Bmn%5E5%7D%7B9%7D+-+3+%5Cfrac%7B1%7D%7B16%7D+m%5E3n+%3D++%5Cfrac%7B1%7D%7B9%7D+%2Amn%5E5++-++%5Cfrac%7B49%7D%7B9%7D+m%5E3n+%3D+mn%28+%5Cfrac%7B1%7D%7B9%7D+n%5E4++-++%5Cfrac%7B49%7D%7B9%7D+m%5E2%29+%3D++%5C%5C++%5C%5C+%0A%3Dmn%28+%28+%5Cfrac%7B1%7D%7B3%7D+n%5E2%29%5E2++-+%28+%5Cfrac%7B7%7D%7B3%7D+m%29%5E2+%29+%3D+mn%28+%5Cfrac%7B1%7D%7B3%7Dn%5E2+-+2+%5Cfrac%7B1%7D%7B3%7Dm%29%28+%5Cfrac%7B1%7D%7B3%7Dn%5E2+%2B2+%5Cfrac%7B1%7D%7B3%7Dm%29)
34.5
5a²+10ab +5b² = 5(a²+2ab+b²)= 5(a+b)²= 5(a+b)(a+b)
2x²+4x+2 = 2(x²+2x+1) = 2(x+1)²= 2(x+1)(x+1)
3m²+3n²-6mn = 3(m² +n² -2mn) = 3(m² -2mn+n²)=
=3(m-n)²=3(m-n)(m-n)
8n² - 16n + 8 = 8(n² - 2n +1) = 8(n-1)² = 8(n-1)(n-1)
34.6
-3x²+12x-12= -3(x² -4x +4) = -3(x² -2*x*2+2²) = -3(x-2)²=-3(x-2)(x-2)
-2a³+20a²b -50ab²= -2a(a² -10ab+ 25b²) = -2a(a²-2*a*5b +(5b)²)=
=-2a(a-5b)² = -2a(a-5b)(a-5b)
-5p²-10pq -5q²= -5(p²+10pq+q²) = -5(p+q)²= -5(p+q)(p+q)
-36z³-24z²-4z = -4z(9z² + 6z +1)= -4z( (3a)² +2*3a*1 +1²)=
= -4z(3a+1)² = -4z(3a+1)(3a+1)