Question

Subject: Factorization 1. Factorise: (a) 14m5n4p² - 42m7n3p⁷ - 70m6n4p³ (b) 2a²(b² - c²) + b² (2c² - 2a²) + 2c²(a² - b²) 2. Express the following as in the form of (a + b)(a - b): (i) a² - 64 (ii) 20a² - 45b² (iii) 32x²y² - 8 (iv) x² - 2xy + y² - z² (v) 49x² - 1 Note: Please explain how you got the answer. Don't spam or else reported.​

Answer 1

Answer:

Solution:

1) Factorise:

a) 14m5n4p² - 42m7n3p⁷ - 70m6n4p³

In this expression 7mnp² is the common factor. This is explained as below: 14m5n4p² = 7 x 2 x m x 5 x n x 4 x p x p

- 42m7n3p⁷ = 7 x -6 x m x 7 x n x 3 x p x p x p x p x p x p x p

- 70m6n4p³ = 7 x -10 x m x 6 x n x 4 x p x p x p

The bolded factors are the common factors in each term and hence we have the common factor as 7mnp².

Taking out 7mnp² from each term, we are left with:

14m5n4p² ------ 40

- 42m7n3p⁷ ------ -126p⁵

-70m6n4p³ ------ -240p

So, the expression becomes:

7mnp² (40 - 126p⁵ - 240p)

Now we can factorise the terms in the brackets to find more common factors:

40 ------ 2 x 2 x 2 x 5

-126p⁵ ------ 2 x 3 x 3 x -7 x p x p x p x p x p

-240p ------ 2 x 2 x 2 x 2 x 3 x -5 x p

Here we can see that the only common factor is 2. Multiply that to 7mnp², we get 14mnp². Taking 14mnp² out of the expression:

= 14mnp² (20 - 63p⁵ - 120p)

b) 2a²(b² - c²) + b² (2c² - 2a²) + 2c²(a² - b²)

First of all, we need to open the brackets. Doing that, we get:

= 2a²b² - 2a²c² + 2b²c² - 2a²b² + 2a²c² - 2b²c²

Cancelling the like terms with different signs:

= 0

2. Express the following as in the form of (a + b)(a - b):

The algebraic identity a² - b² = (a + b)(a - b) is used here.

i) a² - 64

= a² - 8²

= (a + 8)(a - 8)

ii) 20a² - 45b²

= 5 [(4a²) - (9b²)]

= 5 [(2a + 3b) (2a - 3b)]

iii) 32x²y² - 8

= 2 (16x²y² - 4)

= 2 [(4xy)² - (2²)]

= 2 [(4xy + 2) (4xy - 2)]

iv) x² - 2xy + y² - z²

We know that (a - b)² = a² - 2ab + b²

So x² - 2xy + y²

= (x - y)²

Now,

(x - y)² - z²

= x² - y² - z²

= (x -  y + z) (x - y - z)

v) 49x² - 1

= (7x)² - 1²

= (7x + 1) (7x - 1)

Final Answers:

1. a) 14mnp² (20 - 63p⁵ - 120p)

b) 0

2. i) (a + 8)(a - 8)

ii) 5 [(2a + 3b) (2a - 3b)]

iii) 2 [(4xy + 2) (4xy - 2)]

iv) (x -  y + z) (x - y - z)

v) (7x + 1) (7x - 1)

Answer 2

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