factoride the following expressionado
816² 72 bc + 16²
Answers
Answer ♡
Given
Perimeter of Rectangle is 60m
length is 4m more than four times it's breadth
To findi
length and breadth of the Rectangle
Slove
Let us assume the breadth of Rectangle be 'x'
according to the question
length would be => 4m+4x
Perimeter is 60m
Perimeter of Rectangle=2(length+Breadth)
=>60= 2( 4+4x +x)
=>60=2(4+5x)
=>2(4+5x)=60
=>4+5x=30
=>5x= 30-4
=>5x= 26
=>x= 26÷ 5
=>x=5.2
Now , breadth is 5.2m
length = 4+4x=4+4×5.2
length= 4+20.8=24.8m
Check
Perimeter of Rectangle=2( 24.8+5.2)
Perimeter of Rectangle=2(30)
Perimeter of Rectangle= 60m
Extra information=>
Perimeter is the total distance occupy by a solid 2D figure around its edge.
Area of Rectangle= length × breadth
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Step-by-step explanation:
Question 1
Factorize the following expressions.
(i) a² + 8a + 16
(ii) p² – 10 p + 25
(iii) 25m² + 30m + 9
(iv) 49y² + 84yz + 36z²
(v) 4x² – 8x + 4
(vi) 121b² – 88bc + 16c²
(vii) (l + m) ² – 4lm
(viii) a4 + 2a²b² + b4
Answer
We have to make use of following identities to factorize them
(a+b)2= a2 +b2 +2 ab
(a-b)2= a2 +b2 -2 ab
a2 –b2 = (a-b)(a+b)
1) a² + 8a + 16
= a2 + 2×a× 4 + 42
So from first identity, it can be written as
=(a+4)2
2) p² – 10 p + 25
= p2 - 2×p× 5 + 52
So from second identity, it can be written as
=(p-5)2
3) 25m² + 30m + 9
= (5m)2 + 2×5m× 3 + 32
So from first identity, it can be written as
=(5m+3)2
4) 49y² + 84yz + 36z²
= (7y)2 + 2×7y× 6z + (6z)2
So from first identity, it can be written as
=(7y+6z)2
5) 4x² – 8x + 4
= (2x)2 - 2×2x× 2 + 22
So from second identity, it can be written as
=(2x-2)2
= 4(x-1)2 taking common factor 2 out of square
6) 121b² – 88bc + 16c²
= (11b)2 - 2×11b× 4c + (4c)2
So from second identity, it can be written as
=(11b-4c)2
7) (l + m) ² – 4lm
=l2 + m2 +2lm -4lm
= l2 + m2 -2lm
So from second identity, it can be written as
=(l-m)2
8) a4 + 2a²b² + b4
= (a2)2 + 2a²b²+(b2)2
So from first identity, it can be written as
=(a²+b²)²
Question 2
Factorize.
(i) 4p² – 9q²
(ii) 63a² – 112b²
(iii) 49x² – 36
(iv) 16x5 – 144x³
(v) (l + m) ² – (l – m) ²
(vi) 9x² y² – 16
(vii) (x² – 2xy + y²) – z²
(viii) 25a² – 4b² + 28bc – 49c²
Answer
We have to make use of following identities to factorize them
(a+b)2= a2 +b2 +2 ab
(a-b)2= a2 +b2 -2 ab
a2 –b2 = (a-b)(a+b)
1) 4p² – 9q²
=(2p)2 –(3q)2
So from third identity, it can be written as
=(2p-3q)( 2p+3q)
2) 63a² – 112b²
= 7( 9a2 -16b2)
=7[ (3a)2 –(4b)2]
So from third identity, it can be written as
=7(3a-4b)( 3a+4b)
3) 49x² – 36
=(7x)2 –(6)2
So from third identity, it can be written as
=(7x-6)( 7x+6)
4) 16x5-144x3
= x³(16x²-144)
= x³(4x+12)(4x-12)
5) (l + m) ² – (l – m) ²
= [ l+m +l-m][l+m-l+m]
=2l×2m
=4lm
6) 9x² y² – 16
= (3xy-4)(3xy+4)
7) (x² – 2xy + y²) – z²
= (x-y)2 –z2 as (a-b)2= a2 +b2 -2 ab
Now as a2 –b2 = (a-b)(a+b)
= (x-y+z)(x-y-z)
8) 25a² – 4b² + 28bc – 49c²
Factorizing each tem
= (5a)2 –(2b)2 + 2×2b×7c –(7c)2
Rearranging the terms
=(5a)2 –[(2b)2 - 2×2b×7c +(7c)2]
Now as (a-b)2= a2 +b2 -2 ab
=(5a)2 –(2b-7c)2
=(5a-2b+7c)(5a+2b-7c)
Question 3
Factorize the expressions.
(i) ax² + bx
(ii) 7p² + 21q²
(iii) 2x³ + 2xy² + 2xz²
(iv) am² + bm² + bn² + an²
(v) (lm + l) + m + 1
(vi) y (y + z) + 9 (y + z)
(vii) 5y² – 20y – 8z + 2yz
(viii) 10ab + 4a + 5b + 2
(ix) 6xy – 4y + 6 – 9x
Answer
1) ax² + bx
=x(ax+b)
2) 7p² + 21q²
=7(p2+3q2)
3) 2x³ + 2xy² + 2xz²
=2x(x²+y²+z²)
4) am² + bm² + bn² + an²
Rearranging the terms
= am²+ an²+ bm² + bn²
=a(m2 +n2) +b(m2 +n2)
=(a+b) (m2 +n2)
5) (lm + l) + m + 1
=l(m+1) +1(m+1)
=(l+1)(m+1)
6) y (y + z) + 9 (y + z)
= (y+z)(y+9)
7) 5y² – 20y – 8z + 2yz
=5y(y-4) +2z(y-4)
=(5y+2z)(y-4)
8) 10ab + 4a + 5b + 2
=2a(5b+2) +1(5b+2)
=(2a+1)(5b+2)
9) 6xy – 4y + 6 – 9x
=2y(3x-2) +3(2-3x)
=2y(3x-2)-3(3x-2)
=(2y-3)(3x-2)
Question 4
Factorize.
(i) a4 – b4
(ii) p4 – 81
(iii) x4 – (y + z)4
(iv) x4 – (x – z)4
(v) a4 – 2a²b² + b4
Answer:
i) a4-b4 = (a²+b²)(a²-b²)
ii) p4 – 81
=(p²+9)(p²-9)
iii) x4 – (y + z)4
= (x²+(y+z) ²)(x²-(y+z) ²)
= (x²+(y+z) ²)[(x+y+z)(x-y-z)]
iv) x4 – (x – z)4
=(x²-(x-z) ²)(x²+(x-z) ²)
=[(x+x-z)(x-x+z)][(x²+(x-z) ²]
=z(2x-z) [(x²+(x-z) ²]
v) a4 – 2a²b² + b4
=(a2 –b2)2
Question 5
Factorize the following expressions.
(i) p² + 6p + 8
(ii) q² – 10q + 21
(iii) p² + 6p – 16
Answer
1) p²+6p+8
=p(p+6)+8
2) q²-10q+21
=q(q-10)+21
3) p²+6p-16
=p(p+6)-16