confusing math questions?

Question

Factor completely:

81r^4 - 1


and


Simplify:

(5t - 7v)²

Answer 1

81r^4 - 1 is a difference of squares.

x^2 - y^2 factors as (x+y)(x-y), so...

81r^4 - 1 = (9r^2 + 1)(9r^2 - 1)

Note that the second term is itself a difference of squares, so it factors further as:

(9r^2 + 1)(9r^2 - 1) = (9r^2 + 1)(3r + 1)(3r - 1)

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Not sure about "simplifying," but to place (5t - 7v)² in standard form, you'd just multiply it out:

25t^2 - 70vt - 49v^2

Answer 2

Nice factoring problem on the first one... it's a difference of two squares with a li'l surprise.
81r^4 is a perfect square, because 81 is a square number and the exponent is an even number. (9r²)² = 81r^4.
1 is a perfect square, too. 1² = 1.
A difference of two squares factors into a conjugate pair of the roots.
= (9r² + 1)(9r² - 1), and there's the surprise... 9r² - 1 is another difference of two squares. This answer is factored, but not completely so.

(9r² + 1)(9r² - 1)
= (9r² + 1)(3r + 1)(3r - 1). [I think I gave my students this very same problem on a test a few weeks ago!]

In squaring the binomial, please remember that you get three terms... the middle one being twice the product of the terms you're squaring.
(5t - 7v)²
= (5t)² + 2(5t)(-7v) + (-7v)²
= 25t² -70tv + 49v².

Good luck!

Answer 3

1) hint: a^2 - b^2 = (a+b)(a-b) difference of square

81r^4 - 1 = (9r^2+1)(9r^2-1) and the second factor can be factorized using the same rule.

= (9r^2+1)(3r+1)(3r-1)

2) (5t -7v)^2 is just simplified... we can expand it and get

25t^2 -70tv + 49v^2

Answer 4

(9r*2 +1) (9r*2-1) or (9r*2 +1) (3r + 1) (3r -1)

and the second one:

25t*2 -70 tv + 49v*2

Answer 5

(9r^2-1)(9r^2+1)

25t^2 - 70tv + 49v^2

Answer 6

(3r+1)(3r-1)(9r^2+1)