Class 8: Factorization – Exercise 20D

$\displaystyle \textbf{Question 1: }\text{Factorize }x^2+18x+81.$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+18x+81=x^2+2(x)(9)+9^2$
$\displaystyle =(x+9)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorize }a^2-14a+49.$
$\displaystyle \text{Answer:}$
$\displaystyle a^2-14a+49=a^2-2(a)(7)+7^2$
$\displaystyle =(a-7)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorize }4x^2+12x+9.$
$\displaystyle \text{Answer:}$
$\displaystyle 4x^2+12x+9=(2x)^2+2(2x)(3)+3^2$
$\displaystyle =(2x+3)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorize }9x^2-24x+16.$
$\displaystyle \text{Answer:}$
$\displaystyle 9x^2-24x+16=(3x)^2-2(3x)(4)+4^2$
$\displaystyle =(3x-4)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Factorize }49x^2+56x+16.$
$\displaystyle \text{Answer:}$
$\displaystyle 49x^2+56x+16=(7x)^2+2(7x)(4)+4^2$
$\displaystyle =(7x+4)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorize }25z^2-30z+9.$
$\displaystyle \text{Answer:}$
$\displaystyle 25z^2-30z+9=(5z)^2-2(5z)(3)+3^2$
$\displaystyle =(5z-3)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Factorize } 9q^4r^4-6p^4q^2r^2+p^8.$
$\displaystyle \text{Answer:}$
$\displaystyle 9q^4r^4-6p^4q^2r^2+p^8$
$\displaystyle =(3q^2r^2)^2-2(3q^2r^2)(p^4)+(p^4)^2$
$\displaystyle =(3q^2r^2-p^4)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorize } \frac{9p^2}{q^2}+\frac{16r^2}{m^2}+\frac{24pr}{qm}.$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{9p^2}{q^2}+\frac{16r^2}{m^2} +\frac{24pr}{qm}$
$\displaystyle =\left(\frac{3p}{q}\right)^2+ 2\left(\frac{3p}{q}\right)\left(\frac{4r}{m}\right) +\left(\frac{4r}{m}\right)^2$
$\displaystyle =\left(\frac{3p}{q}+\frac{4r}{m}\right)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorize } \frac{1}{4}z^6+9a^2-3az^3.$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{1}{4}z^6+9a^2-3az^3$
$\displaystyle =\left(\frac{1}{2}z^3\right)^2- 2\left(\frac{1}{2}z^3\right)(3a)+(3a)^2$
$\displaystyle =\left(\frac{1}{2}z^3-3a\right)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Factorize } \frac{9}{4}a^2+\frac{49}{9}p^2-7ap.$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{9}{4}a^2+\frac{49}{9}p^2-7ap$
$\displaystyle =\left(\frac{3a}{2}\right)^2- 2\left(\frac{3a}{2}\right)\left(\frac{7p}{3}\right) +\left(\frac{7p}{3}\right)^2$
$\displaystyle =\left(\frac{3a}{2}-\frac{7p}{3}\right)^2.$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Factorize }t^2+22t+85.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Find two numbers whose sum is }22 \text{ and whose product is }85.$
$\displaystyle a+b=22,\qquad ab=85$
$\displaystyle \Rightarrow a=17,\qquad b=5.$
$\displaystyle \therefore t^2+22t+85=(t+17)(t+5).$
$\displaystyle \textit{Note: This technique will be used throughout this exercise.}$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorize }x^2-10x+24.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=-10,\qquad ab=24$
$\displaystyle \Rightarrow a=-6,\qquad b=-4.$
$\displaystyle \therefore x^2-10x+24=(x-6)(x-4).$
$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{Factorize }m^2-3m-40.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=-3,\qquad ab=-40$
$\displaystyle \Rightarrow a=-8,\qquad b=5.$
$\displaystyle \therefore m^2-3m-40=(m-8)(m+5).$
$\displaystyle \\$

$\displaystyle \textbf{Question 14: }\text{Factorize }x^2+x-72.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=1,\qquad ab=-72$
$\displaystyle \Rightarrow a=9,\qquad b=-8.$
$\displaystyle \therefore x^2+x-72=(x+9)(x-8).$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorize }p^2-7p-120.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=-7,\qquad ab=-120$
$\displaystyle \Rightarrow a=8,\qquad b=-15.$
$\displaystyle \therefore p^2-7p-120=(p+8)(p-15).$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{Factorize }16-17z+z^2.$
$\displaystyle \text{Answer:}$
$\displaystyle 16-17z+z^2=z^2-17z+16$
$\displaystyle a+b=-17,\qquad ab=16$
$\displaystyle \Rightarrow a=-1,\qquad b=-16.$
$\displaystyle \therefore 16-17z+z^2=(z-1)(z-16).$
$\displaystyle \\$

$\displaystyle \textbf{Question 17: }\text{Factorize }a^2+5a-104.$
$\displaystyle \text{Answer:}$
$\displaystyle x+y=5,\qquad xy=-104$
$\displaystyle \Rightarrow x=13,\qquad y=-8.$
$\displaystyle \therefore a^2+5a-104=(a+13)(a-8).$
$\displaystyle \\$

$\displaystyle \textbf{Question 18: }\text{Factorize }3x^2+11x+10.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=11,\qquad ab=3\times10=30$
$\displaystyle \Rightarrow a=5,\qquad b=6.$
$\displaystyle 3x^2+11x+10=3x^2+6x+5x+10$
$\displaystyle =3x(x+2)+5(x+2)$
$\displaystyle =(3x+5)(x+2).$
$\displaystyle \\$

$\displaystyle \textbf{Question 19: }\text{Factorize }6x^2+7x-3.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=7,\qquad ab=6\times(-3)=-18$
$\displaystyle \Rightarrow a=9,\qquad b=-2.$
$\displaystyle 6x^2+7x-3=6x^2+9x-2x-3$
$\displaystyle =3x(2x+3)-(2x+3)$
$\displaystyle =(3x-1)(2x+3).$
$\displaystyle \\$

$\displaystyle \textbf{Question 20: }\text{Factorize }3z^2-4z-4.$
$\displaystyle \text{Answer:}$
$\displaystyle a+b=-4,\qquad ab=3\times(-4)=-12$
$\displaystyle \Rightarrow a=-6,\qquad b=2.$
$\displaystyle 3z^2-4z-4=3z^2-6z+2z-4$
$\displaystyle =3z(z-2)+2(z-2)$
$\displaystyle =(z-2)(3z+2).$
$\displaystyle \\$

$\displaystyle \textbf{Question 21: }\text{Factorize }72-x-x^2.$
$\displaystyle \text{Answer:}$
$\displaystyle 72-x-x^2=-x^2-x+72$
$\displaystyle =-(x^2+x-72)$
$\displaystyle =-(x+9)(x-8)$
$\displaystyle =(x+9)(8-x).$
$\displaystyle \\$

$\displaystyle \textbf{Question 22: }\text{Factorize } x^2-3xy-40y^2.$
$\displaystyle \text{Answer:}$
$\displaystyle x^2-3xy-40y^2$
$\displaystyle =x^2-8xy+5xy-40y^2$
$\displaystyle =x(x-8y)+5y(x-8y)$
$\displaystyle =(x+5y)(x-8y).$
$\displaystyle \\$

$\displaystyle \textbf{Question 23: }\text{Factorize } 3x^2y+11xy+6y.$
$\displaystyle \text{Answer:}$
$\displaystyle 3x^2y+11xy+6y=y(3x^2+11x+6)$
$\displaystyle a+b=11,\qquad ab=3\times6=18$
$\displaystyle \Rightarrow a=9,\qquad b=2.$
$\displaystyle y(3x^2+11x+6)=y(3x^2+9x+2x+6)$
$\displaystyle =y\{3x(x+3)+2(x+3)\}$
$\displaystyle =y(x+3)(3x+2).$
$\displaystyle \\$

$\displaystyle \textbf{Question 24: }\text{Factorize } (a-b)^2-5(a-b)+6.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=a-b.$
$\displaystyle \therefore (a-b)^2-5(a-b)+6=x^2-5x+6$
$\displaystyle =(x-2)(x-3).$
$\displaystyle \text{Substituting back,}$
$\displaystyle (a-b)^2-5(a-b)+6=(a-b-2)(a-b-3).$
$\displaystyle \\$

$\displaystyle \textbf{Question 25: }\text{Factorize } (a-3b)^2-4(a-3b)-21.$
$\displaystyle \text{Answer:}$
$\displaystyle (a-3b)^2-4(a-3b)-21$
$\displaystyle =(a-3b)^2-7(a-3b)+3(a-3b)-21$
$\displaystyle =(a-3b)\{(a-3b)-7\}+3\{(a-3b)-7\}$
$\displaystyle =\{(a-3b)+3\}\{(a-3b)-7\}$
$\displaystyle =(a-3b+3)(a-3b-7).$
$\displaystyle \\$

$\displaystyle \textbf{Question 26: }\text{Factorize } 3(y-2)^2-(y-2)-44.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=y-2.$
$\displaystyle \therefore 3(y-2)^2-(y-2)-44=3x^2-x-44$
$\displaystyle =3x^2-12x+11x-44$
$\displaystyle =3x(x-4)+11(x-4)$
$\displaystyle =(3x+11)(x-4).$
$\displaystyle \text{Substituting back,}$
$\displaystyle =\{3(y-2)+11\}\{(y-2)-4\}$
$\displaystyle =(3y+5)(y-6).$
$\displaystyle \\$

$\displaystyle \textbf{Question 27: }\text{Factorize } 7+10(x+y)-8(x+y)^2.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }a=x+y.$
$\displaystyle \therefore 7+10(x+y)-8(x+y)^2=7+10a-8a^2$
$\displaystyle =-8a^2+10a+7$
$\displaystyle =-8a^2+14a-4a+7$
$\displaystyle =-2a(4a-7)-(4a-7)$
$\displaystyle =(7-4a)(2a+1).$
$\displaystyle \text{Substituting back,}$
$\displaystyle 7+10(x+y)-8(x+y)^2$
$\displaystyle =\{7-4(x+y)\}\{2(x+y)+1\}$
$\displaystyle =(7-4x-4y)(2x+2y+1).$
$\displaystyle \\$