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Learn to solve the given algebraic expressions using the below formulas.

(i) a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)

(ii) a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b)

## Factoring Perfect Square Trinomials Examples

1. Factorization when the given expression is a perfect square

(i) m^{4} – 10m^{2}n^{2} + 25n^{4}

Solution:

Given expression is m^{4} – 10m^{2}n^{2} + 25n^{4}

The given expression m^{4} – 10m^{2}n^{2} + 25n^{4} is in the form a^{2} – 2ab + b^{2}.

So find the factors of given expression using a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b) where a = m^{2}, b = 5n^{2}

Apply the formula and substitute the a and b values.

m^{4} – 10m^{2}n^{2} + 25n^{4}

(m^{2})^{2} – 2 (m^{2}) (5n^{2}) + (5n^{2})^{2
}(m^{2} – 5n^{2})^{2}

(m^{2} – 5n^{2}) (m^{2} – 5n^{2})

Factors of the m^{4} – 10m^{2}n^{2} + 25n^{4} are (m^{2} – 5n^{2}) (m^{2} – 5n^{2})

(ii) b^{2}+ 6b + 9

Solution:

Given expression is b^{2}+ 6b + 9

The given expression b^{2}+ 6b + 9 is in the form a^{2} + 2ab + b^{2}.

So find the factors of given expression using a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b) where a = b, b = 3

Apply the formula and substitute the a and b values.

b^{2}+ 6b + 9

(b)^{2} + 2 (b) (3) + (3)^{2
}(b + 3)^{2}

(b + 3) (b + 3)

Factors of the b^{2}+ 6b + 9 are (b + 3) (b + 3)

(iii) p^{4} – 2p^{2} q^{2} + q^{4}

Solution:

Given expression is p^{4} – 2p^{2} q^{2} + q^{4}

The given expression p^{4} – 2p^{2} q^{2} + q^{4} is in the form a^{2} – 2ab + b^{2}.

So find the factors of given expression using a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b) where a = p^{2}, b = q^{2}

Apply the formula and substitute the a and b values.

p^{4} – 2p^{2} q^{2} + q^{4}

(p^{2})^{2} – 2 (p^{2}) (q^{2}) + (q^{2})^{2
}(p^{2} – q^{2})^{2}

(p^{2} – q^{2}) (p^{2} – q^{2})

From the formula (a^{2} – b^{2}) = (a + b) (a – b), rewrite the above equation.

(p + q) (p – q) (p + q) (p – q)

Factors of the p^{4} – 2p^{2} q^{2} + q^{4} are (p + q) (p – q) (p + q) (p – q)

2. Factor using the identity

(i) 25 – a^{2} – 2ab – b^{2}

Solution:

Given expression is 25 – a^{2} – 2ab – b^{2}

Rearrange the given expression as 25 – (a^{2} + 2ab + b^{2})

a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)

25 – (a + b)^{2}

(5)^{2}– (a + b)^{2
}From the formula (a^{2} – b^{2}) = (a + b) (a – b), rewrite the above equation.

[(5 + a + b)(5 – a – b)]

(ii) 1- 2mn – (m^{2} + n^{2})

Solution:

Given expression is 1- 2mn – (m^{2} + n^{2})

1- 2mn – m^{2} – n^{2}

1 – (2mn + m^{2} + n^{2})

1 – (m + n)^{2}

(1)^{2} – (m + n)^{2}

From the formula (a^{2} – b^{2}) = (a + b) (a – b), rewrite the above equation.

(1 + m + n) (1 – m + n)