(1) Factorize the following:
(i) x^{2} + 10x + 24
(ii) z^{2} + 4z - 12
(iii) p^{2} - 6p - 16
(iv) t^{2} + 72 - 17t
(v) y^{2} - 16y - 80
(vi) a^{2} + 10a - 600 Solution
i) (x + 4) and (x + 6)
ii) (z - 2) and (z + 6)
iii) (p - 8) and (p + 2)
iv) (t - 8) and (t - 9)
v) (y - 20) and (y + 4)
vi) (a + 30) and (a - 20)
Factor each trinomial by splitting the middle term.
i) (p - q)^{2} - 6(p - q) - 16
ii) m^{2} + 2mn - 24n^{2}
(iii) √5 a^{2} + 2a - 3√5
(iv) a^{4} - 3a^{2} + 2
(v) 8m^{3} - 2m^{2}n - 15mn^{2}
(vi) (1/x^{2}) + (1/y^{2}) + (2/xy) Solution
i) (p - q + 2) and (p - q - 8).
ii) (m - 4n) and (m + 6n).
iii) (√5a - 3) and (a + √5).
iv) (a + 1), (a - 1) and (a^{2} - 2).
v) (4m + 5n) and (2m - 3n)
vi) (1/x + 1/y) and (1/x + 1/y)
Factorize the following cubic polynomials :
(1) x^{3} - 2x^{2} - 5 x + 6
(2) 4x^{3} - 7x + 3
(3) x^{3 }- 23x^{2 }+ 142x - 120
(4) 4x^{3 }- 5x^{2 }+ 7x - 6
(5) x^{3} - 7x + 6
(6) x^{3} + 13x^{2 }+ 32x + 20 Solution
1) (x - 3)(x + 2)
2) (x - 1)(2x - 1)(2x + 3)
3) (x - 1)(x - 10)(x - 12)
4) (x - 1)(4x^{2 }- x + 6)
5) (x - 1)(x + 3)(x - 2)
6) (x + 1)(x + 10)(x + 2)
Find least common multiple by factoring :
(1) 4x^{2}y, 8x^{3}y^{2}
(2) -9a^{3}b^{2}, 12a^{2}b^{2}c
(3) 16m, -12m^{2}n^{2}, 8n^{2}
(4) p^{2} − 3p +2, p^{2} - 4
(5) 2x^{2} - 5x -3, 4x^{2} -36
(6) (2x^{2} -3xy)^{2}, (4x -6y)^{3}, 8x^{3} -27y^{3}
1) 8 x^{3 }y^{2}
2) 12a^{2} b^{2}c
3) 16m^{2}n^{2}
4) (p + 2)(p - 2)(p - 1)
5) 4(2x + 1)(x + 3)(x - 3)
6) 8x^{2}(2x -3y)^{3}(4x^{2} - 6xy + 9y^{2})
Factor the following polynomial :
(1) 4x + 8
(2) 16a + 64b - 4c
(3) 36x - 16
(4) 35 + 21a
(5) 4a - 8b + 5ax - 10bx Solution
1) 4(x + 2)
2) 4(4a + 16b - c)
3) 4(9x - 4)
4) 7(5 + 3a)
5) (a - 2b) (4 + 5x)
Problem 1 :
The sum of a number and its reciprocal is 65/8. Find the number
Problem 2 :
The difference of the squares of two positive numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.
Problem 3 :
A farmer wishes to start a 100 sq.m rectangular vegetable garden. Since he has only 30 m barbed wire, he fences the sides of the rectangular garden letting his house compound wall acts the fourth side fence. Find the dimension of the garden.
Problem 4 :
A rectangular field is 20 m long and 14 m wide. There is the path of equal width all around it having an area of 111 sq.m. Find the width of the path on the outside.
Problem 5 :
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train
1) So, the required number is 8.
2) Therefore the required numbers are 9 and 6.
3)
length of garden = 10 m and
width of the garden = 5 m.
4) Therefore width of the path = 1.5 m.
5) Therefore speed of the train is 45 km/hr.
Problem 1 :
If the difference between a number and its reciprocal is ²⁴⁄₅, find the number.
Problem 2 :
A garden measuring 12m by 16m is to have a pedestrian pathway that is w meters wide installed all the way around so that it increases the total area to 285 m^{2}. What is the width of the pathway?
Problem 3 :
A bus covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hr more, it would have taken 30 minutes less for the journey. Find the original speed of the bus.
Problem 4 :
John and Jivanti together had 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they have now is 124. Find the number marbles each one them had initially.
Problem 5 :
A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in dollars) was found to be 55 minus the number of toys produced in a day, the total cost of production was $750. Find the number of toys produced on that day.
Problem 6 :
The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.
Problem 7 :
The difference of squares of two positive numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Problem 8 :
A train travels 360 miles at a uniform speed. If the speed had been 5 miles/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.
Problem 9 :
Find two consecutive positive even integers whose squares have the sum 340.
Problem 10 :
The sum of squares of three consecutive natural numbers is 194. Determine the numbers.
1) The required numbers are 5 and 1/5
2) the width of the pathway is 1.5 m.
3) the original speed of the bus is 45 km per hour.
4) So, John had 36 marbles, when Jivanti had 9 marbles or Jivanti had 36 marbles, when John had 9 marbles.
5) the number of toys produced on that particular day is 30 or 25.
6)
length of the shorter side = 90 m
length of the longer side = 90 + 30 = 120 m
7) the larger number is 18 and the smaller number is 12.
8) the original speed of the train is 40 miles/hr.
9) the two positive even integers are 12 and 14.
10) the three consecutive natural numbers are 7, 8 and 9.
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