Factorize xsq+25ysq+81zsq+10xy+90yz+18zx
Answers
Answer:
3y(5y-7)
Step-by-step explanation:
(i) y^2+7y+10\ =\ y^2+2y+5y+10y
2
+7y+10 = y
2
+2y+5y+10
=\ y\ \left(y+2\right)+5\left(y+2\right)= y (y+2)+5(y+2)
=\ \left(y+5\right)\left(y+2\right)= (y+5)(y+2)
Division: \frac{\left(y+5\right)\left(y+2\right)}{y+5}=\ y+2
y+5
(y+5)(y+2)
= y+2
(ii) m^2-14m-32\ =\ m^2-16m+2m-32m
2
−14m−32 = m
2
−16m+2m−32
=\ m\left(m-16\right)+2\left(m-16\right)= m(m−16)+2(m−16)
=\ \left(m+2\right)\left(m-16\right)= (m+2)(m−16)
Division: \frac{\left(m+2\right)\left(m-16\right)}{m+2}
m+2
(m+2)(m−16)
= m-16
(iii) 5p^2-25p+20\ =\ 5\left(p^2-5p+4\right)5p
2
−25p+20 = 5(p
2
−5p+4)
= 5\left(p^2-4p-p+4\right)5(p
2
−4p−p+4)
=5\left[p\left(p-4\right)-1\left(p-4\right)\right]5[p(p−4)−1(p−4)]
=5\left(p-1\right)\left(p-4\right)5(p−1)(p−4)
Division: \frac{5\left(p-1\right)\left(p-4\right)}{p-1}
p−1
5(p−1)(p−4)
= 5(p-4)
(iv) 4yz\left(z^2+6z-16\right)=4yz\left[z^2+8z-2z-16\right]4yz(z
2
+6z−16)=4yz[z
2
+8z−2z−16]
=4yz\left[z\left(z+8\right)-2\left(z+8\right)\right]=4yz[z(z+8)−2(z+8)]
4yz\left(z-2\right)\left(z+8\right)4yz(z−2)(z+8)
division = \frac{4yz\left(z-2\right)\left(z+8\right)}{2y\left(z+8\right)}\ =\ 2z\left(z-2\right)
2y(z+8)
4yz(z−2)(z+8)
= 2z(z−2)
(v) 5pq\left(p^2-q^2\right)=5pq\left(p+q\right)\left(p-q\right)5pq(p
2
−q
2
)=5pq(p+q)(p−q)
Division: \frac{5pq\left(p+q\right)\left(p-q\right)}{2p\left(p+q\right)}
2p(p+q)
5pq(p+q)(p−q)
=\ \frac{5}{2}q\left(p-q\right)=
2
5
q(p−q)
(vi) 12xy\left(9x^2-16y^2\right)=\ \ 2\times2\times3\times x\times y\times\left[\left(3x\right)^2-\left(4y\right)^2\right]12xy(9x
2
−16y
2
)= 2×2×3×x×y×[(3x)
2
−(4y)
2
]
=3\times4xy\times\left[\left(3x+4y\right)\left(3x-4y\right)\right]=3×4xy×[(3x+4y)(3x−4y)]
Division: \frac{3\times4xy\times\left(3x+4y\right)\left(3x-4y\right)}{4xy\left(3x+4y\right)}
4xy(3x+4y)
3×4xy×(3x+4y)(3x−4y)
= 3(3x-4y)
(vii) 39y^3\left(50y^2-98\right)=\ 3\times13\times y\times y\times y\times2\left(25y^2-49\right)39y
3
(50y
2
−98)= 3×13×y×y×y×2(25y
2
−49)
=3\times13\times y\times y\times y\times2\left(\left(5y\right)^2-\left(7\right)^2\right)3×13×y×y×y×2((5y)
2
−(7)
2
)
=3\times13\times2\times y\times y\times y\times\left[\left(5y+7\right)\left(5y-7\right)\right]3×13×2×y×y×y×[(5y+7)(5y−7)]
26y^2\left(5y+7\right)=2\times13\times y\times y\times\left(5y+7\right)26y
2
(5y+7)=2×13×y×y×(5y+7)
Division = \frac{2\times3\times13\times y\times y\times y\times\left(5y+7\right)\left(5y-7\right)}{2\times13\times y\times y\times\left(5y+7\right)}
2×13×y×y×(5y+7)
2×3×13×y×y×y×(5y+7)(5y−7)
=\ 3\times y\times\left(5y-7\right)= 3×y×(5y−7)
=3y(5y-7)