Question

Factorise the following expressions. 

1.  \(a^ 2 + 8a + 16 \)
2. \(p^ 2 - 10p + 25 \)
3. \(25m^2 + 30m + 9 \)
4. \(49y^ 2 + 84yz + 36z ^2 \)
5. \(4x^ 2 - 8x + 4 \)
6. \(121b ^2 - 88bc + 16c ^2 \)
7. \((l + m) ^2 - 4lm \) (Hint: Expand \((l + m) ^2 \) first)
8.  \(a^ 4 + 2a ^2b ^2 + b^ 4\)

Answer 1

(i) \(a^ 2 + 8a + 16 = (a)^ 2 + 2 \times a \times 4 + (4)^2 \)
=\( (a + 4)^2 [(x + y)^ 2 = x ^2 + 2xy + y^ 2 ]\)

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(ii) \(p^ 2 - 10p + 25 = (p)^ 2 - 2 \times p \times 5 + (5)^2 \)
= \((p - 5)^2 [(a - b)^ 2 = a ^2 - 2ab + b ^2]\)

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(iii) \(25m^2 + 30m + 9 = (5m) ^2 + 2 \times 5m \times 3 + (3)^2 \)
= \((5m + 3)^2 [(a + b) ^2 = a ^2 + 2ab + b^ 2 ]\)

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(iv) \(49y ^2 + 84yz + 36z^ 2 = (7y)^ 2 + 2 \times (7y) \times (6z) + (6z)^ 2 \)
= \((7y + 6z)^ 2 [(a + b)^ 2 = a ^2 + 2ab + b^ 2 ]\)

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(v) \(4x^ 2 - 8x + 4 = (2x)^ 2 - 2 (2x) (2) + (2)^2 \)
= \((2x - 2)^2 [(a - b)^ 2 = a^ 2 - 2ab + b ^2 ]\)
= \([(2) (x - 1)]^2 = 4(x - 1)^2\)

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(vi) \(121b^ 2 - 88bc + 16c^ 2 = (11b)^ 2 - 2 (11b) (4c) + (4c)^ 2\)

= \((11b - 4c)^ 2 [(a - b)^ 2 = a ^2 - 2ab + b ^2 ]\)

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(vii) \((l + m)^ 2 - 4lm = l^ 2 + 2lm + m^2 - 4lm \)
=\( l^ 2 - 2lm + m^2 \)
= \((l - m)^ 2 [(a - b)^ 2 = a ^2 - 2ab + b^ 2 ]\)

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(viii) \(a^ 4 + 2a ^2b^ 2 + b^ 4 = (a^ 2 )^ 2 + 2 (a ^2 ) (b^ 2 ) + (b^ 2 )^ 2 \)
= \((a^ 2 + b^ 2 )^ 2 [(a + b)^ 2 = a^ 2 + 2ab + b ^2 ]\)