Factorisation Test

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Factorisation Test - 23

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Question 1
1 / -0

Directions For Questions

Divide the given polynomial by the given monomial.

...view full instructions

$$(5a^3 b - 7ab^3) \div ab$$

A
$$5a^2 - 7b^2$$

B
$$5a - 7b^2$$

C
$$5a^2 - b^2$$

D
None

Solution

We divide the given polynomial $$5a^3b-7ab^3$$ by the monomial $$ab$$ as shown below:

$$\dfrac { 5a^{ 3 }b-7ab^{ 3 } }{ ab } \\ =\dfrac { 5a^{ 3 }b }{ ab } -\dfrac { 7ab^{ 3 } }{ ab } \\ =\dfrac { 5a^{ 3 }b }{ ab } -\dfrac { 7ab^{ 3 } }{ ab } \\ =\dfrac { 5\times a\times a\times a\times b }{ a\times b } -\dfrac { 7\times a\times b\times b\times b }{ a\times b } \\ =5a^{ 2 }-7b^{ 2 }$$

Hence, $$\dfrac { 5a^{ 3 }b-7ab^{ 3 } }{ ab } =5a^{ 2 }-7b^{ 2 }$$.
Question 2
1 / -0

Divide the given polynomial by the given monomial:

$$\left(\dfrac{2}{3} a^2 b^2 c^2 + \dfrac{4}{3} ab^2 c^2 \right ) \div \dfrac{1}{2} abc$$

A
$$\dfrac{4}{3} (abc + bc)$$

B
$$\dfrac{2}{3} (abc + 2bc)$$

C
$$\dfrac{4}{3} (abc + 2bc)$$

D
None

Solution

We divide the given polynomial $$\dfrac { 2 }{ 3 } a^{ 2 }b^{ 2 }c^{ 2 }+\dfrac { 4 }{ 3 } ab^{ 2 }c^{ 2 }$$ by the monomial $$\dfrac { 1 }{ 2 } abc$$ as shown below:

$$\dfrac { \dfrac { 2 }{ 3 } a^{ 2 }b^{ 2 }c^{ 2 }+\dfrac { 4 }{ 3 } ab^{ 2 }c^{ 2 } }{ \dfrac { 1 }{ 2 } abc } \\ =\dfrac { \dfrac { 2 }{ 3 } a^{ 2 }b^{ 2 }c^{ 2 } }{ \dfrac { 1 }{ 2 } abc } +\dfrac { \dfrac { 4 }{ 3 } ab^{ 2 }c^{ 2 } }{ \dfrac { 1 }{ 2 } abc }$$
$$=\dfrac { \dfrac { 2 }{ 3 } \times a\times a\times b\times b\times c\times c }{ \dfrac { 1 }{ 2 } \times a\times b\times c } +\dfrac { \dfrac { 4 }{ 3 } \times a\times b\times b\times c\times c }{ \dfrac { 1 }{ 2 } \times a\times b\times c } \\ =\left( \dfrac { 2 }{ 3 } \times 2\times a\times b\times c \right) +\left( \dfrac { 4 }{ 3 } \times 2\times b\times c \right) \\ =\dfrac { 4 }{ 3 } abc +\dfrac { 8 }{ 3 } bc \\ =\dfrac { 4 }{ 3 } (abc+2bc)$$

Hence, $$\dfrac { \dfrac { 2 }{ 3 } a^{ 2 }b^{ 2 }c^{ 2 }+\dfrac { 4 }{ 3 } ab^{ 2 }c^{ 2 } }{ \dfrac { 1 }{ 2 } abc } =\dfrac { 4 }{ 3 } (abc+2bc)$$.
Question 3
1 / -0

Directions For Questions

Divide the given polynomial by the given monomial.

...view full instructions

$$(25x^5 - 15x^4) \div 5x^3$$

A
$$x(5x + 3)$$

B
$$(5x - 3)$$

C
$$x(5x - 3)$$

D
None

Solution

We divide the given polynomial $$25x^5-15x^4$$ by the monomial $$5x^3$$ as shown below:

$$\dfrac { 25x^{ 5 }-15x^{ 4 } }{ 5x^{ 3 } } \\ =\dfrac { 25x^{ 5 } }{ 5x^{ 3 } } -\dfrac { 15x^{ 4 } }{ 5x^{ 3 } } \\ =\left( \dfrac { 5\times 5\times x^{ 5 } }{ 5x^{ 3 } } \right) -\left( \dfrac { 3\times 5\times x^{ 4 } }{ 5x^{ 3 } } \right) \\ =\left( \dfrac { 5\times x^{ 5 }\times x^{ -3 } }{ 1 } \right) -\left( \dfrac { 3\times x^{ 4 }\times x^{ -3 } }{ 1 } \right)$$
$$=(5\times x^{ (5-3) })-(3\times x^{ (4-3) })\quad \quad \quad \quad \quad \quad \left( \because \quad a^{ x }+a^{ y }=a^{ x+y } \right) \\ =5x^{ 2 }-3x\\ =x(5x-3)$$

Hence, $$\dfrac { 25x^{ 5 }-15x^{ 4 } }{ 5x^{ 3 } } =x(5x-3)$$.
Question 4
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Factorize the following
$$pq - pr - 3ps$$

A
$$p(q + r - 3s)$$

B
$$p(q + r + 3s)$$

C
$$q(q + r - 3s)$$

D
None of these

Solution

$$pq-pr-3ps$$

(To factorise means to make factor)

Taking $$p$$ common from all $$3$$ equations, we have

$$p(q-r-3s)$$
Question 5
1 / -0

Factorise the expression:
$$2a^3 - 3a^2 b + 2a^2 c$$

A
$$a^2 (2a - 3b + 2c)$$

B
$$a^2 (2a + 3b + 2c)$$

C
$$a^2 (2a - 3b - 2c)$$

D
None of these

Solution

$$2a^{3}-3a^{2}b+2a^{2}c$$

Taking $$a^{2}$$ common from all $$3$$ terms

$$=a^{2}(2a-3b+2c)$$
Question 6
1 / -0

Factorize the following expression
$$16x + 64x^2 y$$

A
$$16x (1 + 4xy)$$

B
$$6x (1 + 4xy)$$

C
$$16x (1 - 4xy)$$

D
None of these

Solution

$$16x+64x^{2}y$$

$$=16x+4\times (16x)(xy)$$

Taking $$16x$$ common

$$=16x(1+4xy)$$
Question 7
1 / -0

Factorize the following expression:
$$10x^3 - 25 x^4 y$$

A
$$5x^3 (2 - 5xy)$$

B
$$5x^3 (2 + 5xy)$$

C
$$x^3 (2 - 5xy)$$

D
None of these

Solution

Taking $$5x^{3}$$ common from all the terms

$$5x^{3}(2-5xy)$$
Question 8
1 / -0

Find the quotient and remainder of the given expression,
$$(3x^3+4x^2-5)\div (3x+1)$$

A
$$x^2+x-\dfrac{1}{3}, -\dfrac{14}{3}$$

B
$$x^2-3x+\dfrac{5}{9}, \dfrac{14}{3}$$

C
$$x^2+3x+\dfrac{5}{9}, -\dfrac{14}{3}$$

D
None of these

Solution
Question 9
1 / -0

Find the quotient and remainder of the given expression, $$(3x^3+2x^2+7x-5)\div (x+3)$$

A
$$3x^2-7x+28, -89$$

B
$$3x^2-7x+40, 125$$

C
$$3x^2-11x+40, -15$$

D
None of these

Solution

We divide $$3x^3+2x^2+7x-5$$ by $$(x+3)$$ as shown in the above image:

From the division, we observe that the quotient is $$3x^2-7x+28$$ and the remainder is $$-89$$.

Hence, the quotient is $$3x^2-7x+28$$ and the remainder is $$-89$$.
Question 10
1 / -0

Find the quotient and remainder of the given expression,
$$(8x^4-2x^2+6x-5)\div (4x+1)$$

A
$$2x^3-\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, -\dfrac{211}{32}$$

B
$$2x^3-\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, \dfrac{211}{32}$$

C
$$2x^3+\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, -\dfrac{211}{32}$$

D
None of these

Solution

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Re-Attempt Weekly Quiz Competition

Factorisation Test - 23

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Factorisation Test - 23

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Question 1

$$5a^2 - 7b^2$$

$$5a - 7b^2$$

$$5a^2 - b^2$$

None

Solution

Question 2

$$\dfrac{4}{3} (abc + bc)$$

$$\dfrac{2}{3} (abc + 2bc)$$

$$\dfrac{4}{3} (abc + 2bc)$$

None

Solution

Question 3

$$x(5x + 3)$$

$$(5x - 3)$$

$$x(5x - 3)$$

None

Solution

Question 4

$$p(q + r - 3s)$$

$$p(q + r + 3s)$$

$$q(q + r - 3s)$$

None of these

Solution

Question 5

$$a^2 (2a - 3b + 2c)$$

$$a^2 (2a + 3b + 2c)$$

$$a^2 (2a - 3b - 2c)$$

None of these

Solution

Question 6

$$16x (1 + 4xy)$$

$$6x (1 + 4xy)$$

$$16x (1 - 4xy)$$

None of these

Solution

Question 7

$$5x^3 (2 - 5xy)$$

$$5x^3 (2 + 5xy)$$

$$x^3 (2 - 5xy)$$

None of these

Solution

Question 8

$$x^2+x-\dfrac{1}{3}, -\dfrac{14}{3}$$

$$x^2-3x+\dfrac{5}{9}, \dfrac{14}{3}$$

$$x^2+3x+\dfrac{5}{9}, -\dfrac{14}{3}$$

None of these

Solution

Question 9

$$3x^2-7x+28, -89$$

$$3x^2-7x+40, 125$$

$$3x^2-11x+40, -15$$

None of these

Solution

Question 10

$$2x^3-\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, -\dfrac{211}{32}$$

$$2x^3-\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, \dfrac{211}{32}$$

$$2x^3+\dfrac{x^2}{2}-\dfrac{3}{8}x+\dfrac{51}{32}, -\dfrac{211}{32}$$

None of these

Solution

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