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# Write a linear factorization of the function. f(x) = x^{4} + 36x^{2}

**Solution:**

Given function f(x) = x^{4} + 36x^{2}

= (x^{2})^{2} + (6x)^{2}

We know that a^{2} + b^{2} = (a + b)^{2} - 2ab

Therefore,

= (x^{2})^{2} + (6x)^{2}

= (x^{2} + 6x)^{2} - 2(x^{2})(6x)

= x^{2}(x - 6)^{2} - 12x^{3}

= x^{2}[(x - 6)^{2} - 12x]

= x^{2}[(x - 6)^{2} - (√12x)^{2} ]

We know that a^{2} - b^{2} = (a + b)(a - b)

(x - 6)^{2} - (√12x)^{2} = (x - 6 + √12x)(x - 6 - √12x), therefore

x^{2}[(x - 6)^{2} - (√12x)^{2} ]

= x^{2}(x - 6 + √12x)(x - 6 - √12x)

Alternatively

f(x) = x^{4} + 36x^{2}

= x^{2}(x^{2} +36)

= x^{2}(x + 6i)(x - 6i)

The linear factors are x, (x - 6i) and (x + 6i)

## Write a linear factorization of the function. f(x) = x^{4} + 36x^{2}

**Summary:**

Linear factorization of the function. f(x) = x^{4} + 36x^{2} leads to x^{2}(x + 6i)(x - 6i)

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