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Factorization

Combinations

Sometimes, in order to factorise an expression as fully as possible, you need a combination of quadratic factorisation with the taking out of other common factors.

Example 10  
2x2(x + 1) - 4x
Strategy

2x is in common.

2x2(x + 1) - 4x
= 2x(x(x + 1) - 2(2x))
= 2x(x(x + 1) - 2)
= 2x(x2 + x - 2)

Factorise by taking out common factors first.

Try to simplify the expression in the brackets. If it is a quadratic expression, try to factorise it.

For x2 + x - 2, A = 1, B = 1, C = -2 Obtain the values of A, B and C by inspection



Hence x2 + x - 2 = (x - 1)(x + 2)

Substitute the values into the quadratic formula and simplify.

To make the factorisation correct we have to multiply by the coefficient of x2 in the original quadratic, that is multiply by A. (No action is required if A = 1.)

2x2(x + 1) - 4x = 2x(x - 1)(x + 2)

Put it all together.

 

Now try your hand at some really tough examples!

Exercise 10

Factorise each expression as far as possible.

1. 5x4 - 20x2

2. x2ex + 2xex + 5ex

3. y(y + 5)sin 3x + 6sin 3x

4. 3x2(x + 9) - 42x(x + 1)

5. 8x - 2x3

6. x(4 - x)(7 - 2x) + 4(4 - x)

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