factorization

Resolve into factors:

x⁴ +x²y² + y⁴

Answer:

x⁴ +x²y² + y⁴

= X⁴ + 2x²y² + y⁴ - x²y²

=(x² +y²)² - x²y²

=(x² +y² – xy) (x² +y² + xy)

Resolve into factors:

x⁴ - x² + 2x - 1

Answer:

x⁴ - x² + 2x - 1

= x⁴ –( x² - 2x + 1)

= x⁴ – (x - 1)²

=( x² + x – 1)( x² - x + 1)

Resolve into factors:

x² -  7x – 228

Answer:

x² -  7x – 228

= x² - 7x + (7/2)²– 228 -(7/2)²

=(x-7/2)² -961/4

=(x -7/2 -31/2)(x – 7/2 +31/2)

=(x +12)(x -19)

harder factorisation

Resolve into factors:

15 y² + 22y – 48

Answer:

Multiply and divide the expression by 15.

So,  ¹/₁₅ {(15y)² + 22.15y – 48.15}

Let 15y = x

Therefore, 1/15 {x² + 22x - 720}

=¹/₁₅{x² + 40x – 18x - 720}

=¹/₁₅(x+40)(x-18)

=¹/₁₅(15y +40)(15y- 18)  substituting x = 15y

= (3y +8)(5y -6)

Resolve into factors:

28 + 31p -5p²

Answer:

28 + 31p -5p²

=28 + 35p – 4p - 5p²

=7(4 + 5p) –p(4 + 5p)

= (7 – p)(4 + 5p)

 

Resolve into factors:

15 + 16p – 15p²

Answer:

15 + 16p – 15p²

Multiply and divide the numerator and denominator by 15

= ¹/₁₅{15.15 + 15.16p – (15p)²}

Let, 15p = y

= -¹/₁₅{y2 – 16y +15} 

=-¹/₁₅{y2 – 15y - y +15} 

=-¹/₁₅{y(y – 15) -1( y -15)} 

= -¹/₁₅(y -1)(y – 15)

= -¹/₁₅(15p -1)(15p -15)

= - (15p -1)(p – 1)

= (15p -1)(1- p)

factorization

Resolve in factors:

x⁴ – 10 x² + 9

Answer

Let x² = y

So, we have y² – 10y + 9

= y² – 9y – y + 9

= y(y – 9) – 1(y – 9)

= (y – 1)(y – 9)

= (x² – 1)( x²– 9)

=(x-1)(x +1)(x -9)(x +9)

Resolve into factors:

x² + 2xy + y² – x – y

Answer:

x² + 2xy + y² – x – y

= (x + y)² – (x – y)

=(x + y)(x + y – 1)

Resolve into factors:

 a(a+1) – b(b+ 1)

Answer:

= a² + a – b² – b

= (a – b)(a + b) +(a – b)

=(a – b) (a + b + 1)

Barter

A man has bought 6 oxen and 5 cow of $ 17120 and would lose $160 by exchanging 3 oxen for 5 cows. What is the price of a cow?

Answer:

Exchange: 3 oxen for 5 cows

So, 6 oxen for 10 cows == loss = 160 x 2 = $320

Now, 6 oxen + 5 cow = 10 cow + 5 cow = $17120 – $320 = $16800

Therefore, 15 cow = $ 16800

Or, 1 cow = 16800/15 = $ 1120

If 50 meters of muslin and $ 360 is given in exchange for 36 meters of silk at $30 per meter, then what is the price of muslin?

Answer:

Cost of 36 meters of silk = $30 x 36 = $ 1080

50 meters of muslin + $ 360 = $ 1080

Therefore 50 meters of muslin = $720

So, price of muslin = $720/50 = $ 14.4

mixtures, STD V

A, B and C have $ 507 between them. B and C together have $ 345 and C and A together have $ 305. How much does C have?

Answer:

A+B+C =507--------(1)

B+C = 345----------(2)

A+C = 305----------(3)

Adding (2) and (3), we get A+2B+C = 345 + 305------(4)

Substituting (1) in (4), we get

507 + B = 650

Or, B = 143

Therefore, C = 345 – 143 = 202

A Sum of money is divided into A, B and C. C gets twice as much as A, A and B together get $70, and B and C together get $90. How much does each person get?

Answer:

2A = C -------(1)

A + B = 70------(2)

B +C = 90-------(3)

(2) – (3)

A – C = -20,-----(4)

Substituting (1) in (4), we get

A – 2A = -20

Or, A = 20, therefore B = 50 and C = 40

Tougher questions in LCM

Find the greatest number less than 900 which is exactly divisible by 8, 12 and 28?

Answer:
LCM of 8,12, 28 = 168
to find the multiple of 168 = 900/168 =  5 60/168.
So, greatest number less than 900 which is exactly divisible = 900 - 60 = 840.

What greatest number can be subtracted from 2470 so that the Remainder may be divisible by 42, 98 and 105?

Answer:
Remainder will be LCM of 42, 98, 105 = 1470
So, the greatest number that can be subtracted = 2470 - 1470 = 1000.

Find the least number that being increased by 8 is divisible by 21, 35 and 48?

Answer:
LCM of 21, 35, 48 = 1680.
SO, the least number that being increased by 8 = 1680 -8 = 1672.

Find the least number which when divided by 18, 24, 30 and 42 will leave a remainder of 1 in each case.

Answer:
LCM of 18, 24, 30, 42 = 2520 
So, the least number that would leave a remainder of 1 = 2520 + 1 = 2521.

Tougher questions in division

A number when divided by 11 has a remainder 8 and the quotient divided by 13 has a remainder 7. What will be the respective remainders when the order of the divisor is reversed?

What is the nearest integer to 56100 which is exactly divisible by 456?

56100 / 456 = 123 12/456
so, the nearest integer = 56100 - 12 = 56,088

A boy had to divide 49471 by 210. Instead he divides it by some other number and gets a quotient of 246 and a remainder of 25. What was the divisor?

A number when divided by 221 gave a remainder of 43. What will be the remainder if the same number is divided by 17?

4767 exactly divides the number ***341. Find the missing digits indicated by the star.

Answer:
4767 x 3 =            14301
4767 x 2 =            9534x
4767 x 1 =          4767xx
----------------------------------
                           586,341
the missing digits are 586.

Two numbers when divided  by a certain divisor, leaves a remainder 4375 and 2986 respectively. When the sum of the two numbers is divided by the same divisor, the remainder is 2361. What is the divisor?
Answer:
Divisor = 4375 + 2986 - 2361 = 5000

A number when divided by 8,9 and 10 successively ( by the method of factor) leaves Remainder of 3,4 and 5 respectively. What will be the Remainder if the divisors were 3,5, 6 and 8 in order?