GCSE Edexcel Maths Geometry Toolkit: 14 (a) Factorise fully 4p²

14 (a) Factorise fully 4p² - 36 (b) Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d where a, b, c and d are integers. - Edexcel - GCSE Maths - Question 15 - 2022 - Paper 3

Question 15

14 (a) Factorise fully 4p² - 36 (b) Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d where a, b, c and d are integers.

Worked Solution & Example Answer:14 (a) Factorise fully 4p² - 36 (b) Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d where a, b, c and d are integers. - Edexcel - GCSE Maths - Question 15 - 2022 - Paper 3

Step 1

Factorise fully 4p² - 36

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Answer

To factorise the expression 4p² - 36, we start by recognizing that it is a difference of squares. We can express this as:

$4p^2 - 36 = (2p)^2 - 6^2$

According to the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$ , we can rewrite the expression as:

$(2p + 6)(2p - 6)$

Next, we can factor out a common factor from each of the terms:

$2(p + 3)(p - 3)$

So, the fully factored form of 4p² - 36 is:

$2(p + 3)(p - 3)$

Step 2

Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d

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Answer

To show that the expression (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d, we will first expand the expression step by step.

Start by expanding (m + 4)(2m - 5):

$(m + 4)(2m - 5) = m(2m) + m(-5) + 4(2m) + 4(-5)$

$= 2m^2 - 5m + 8m - 20$

$= 2m^2 + 3m - 20$
Next, we will multiply this result by (3m + 1):

$(2m^2 + 3m - 20)(3m + 1)$

Expanding gives:

$= 2m^2(3m) + 2m^2(1) + 3m(3m) + 3m(1) - 20(3m) - 20(1)$

$= 6m^3 + 2m^2 + 9m^2 + 3m - 60m - 20$

$= 6m^3 + (2m^2 + 9m^2) + (3m - 60m) - 20$

$= 6m^3 + 11m^2 - 57m - 20$

Thus, we have expressed (m + 4)(2m - 5)(3m + 1) as:

$6m^3 + 11m^2 - 57m - 20$

This verifies that it can indeed be arranged in the desired form where all constants are integers, specifically where:

a = 6
b = 11
c = -57
d = -20.

14 (a) Factorise fully 4p² - 36 (b) Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d where a, b, c and d are integers. - Edexcel - GCSE Maths - Question 15 - 2022 - Paper 3

Worked Solution & Example Answer:14 (a) Factorise fully 4p² - 36 (b) Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d where a, b, c and d are integers. - Edexcel - GCSE Maths - Question 15 - 2022 - Paper 3

Factorise fully 4p² - 36

Show that (m + 4)(2m - 5)(3m + 1) can be written in the form am² + bm + cm + d

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