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How do you factor #16ax^3 + 54ay^3#?

Answer 1

Avery Alexander

#16ax^3+54ay^3#

#=2a(8x^3+27y^3)#

#=2a((2x)^3+(3y)^3)#

#=2a(2x+3y)((2x)^2-(2x)(3y)+(3y)^2)#

#=2a(2x+3y)(4x^2-6xy+9y^2)#

...using the identity of the sum of cubes:

#A^3+B^3 = (A+B)(A^2-AB+B^2)#

Simpler factors with real coefficients do not exist.

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Answer 2

Avery Alexander

To factor (16ax^3 + 54ay^3), first find the greatest common factor (GCF) of the two terms. The GCF of (16ax^3) and (54ay^3) is (2a). Then, factor out (2a) from both terms. After factoring out the GCF, you are left with (2a(8x^3 + 27y^3)).

Now, observe that (8x^3) and (27y^3) are both perfect cubes. Apply the formula for factoring the sum of cubes, which is (a^3 + b^3 = (a + b)(a^2 - ab + b^2)).

Using this formula, we can factor (8x^3 + 27y^3) as ((2x)^3 + (3y)^3), where (a = 2x) and (b = 3y).

So, (8x^3 + 27y^3 = (2x + 3y)(4x^2 - 6xy + 9y^2)).

Therefore, the fully factored form of (16ax^3 + 54ay^3) is (2a(2x + 3y)(4x^2 - 6xy + 9y^2)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.