How do you know if #5y^4+10y^2+5# is a perfect square trinomial and how do you factor it?

Answer 1

It is a perfect square trinomial, it is of the form

#a^2+2ab+b^2 = (a+b)^2#, with #a=sqrt(5)y^2# and #b=sqrt(5)#.

So we can write:

#5y^4+10y^2+5 = (sqrt(5)y^2+sqrt(5))^2#

Generally it's nicer to use rational coefficients if possible and write instead:

#5y^4+10y^2+5 = 5(y^2+1)^2#
#(y^2+1)# has no simpler factors with real coefficients because:
#y^2+1 >= 1 > 0# for all #y in RR#
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Answer 2

To determine if the trinomial 5y^4 + 10y^2 + 5 is a perfect square trinomial, you need to check if the first and last terms are perfect squares, and if the middle term is twice the product of the square roots of the first and last terms.

In this case, the first and last terms are 5y^4 and 5, respectively. Both of these terms are perfect squares since they can be expressed as (y^2)^2 and (1)^2, respectively.

Now, find the square root of the first term: √(5y^4) = √(5y^2)^2 = 5y^2. Then, find the square root of the last term: √(5) = √(1)^2 = 1.

Now, check if the middle term, 10y^2, is twice the product of the square roots of the first and last terms: 2 * (5y^2) * (1) = 10y^2.

Since the middle term is indeed twice the product of the square roots of the first and last terms, the trinomial 5y^4 + 10y^2 + 5 is a perfect square trinomial.

To factor a perfect square trinomial, you take the square root of the first and last terms and write it as a binomial squared. In this case:

5y^4 + 10y^2 + 5 = (5y^2 + 1)^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.

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.

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.

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.

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