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Question 176132: Please help!
1. Using one of the laws of exponents, prove that nay number raised to the power 0 is 1.
2. Using FOIL, simplify the expression (3x+2)(3x-2). Show that a particular factoring formula leads to the same answer.
3. If a fourths-sdegree polynomial is multiplied by third-degree polnomial, what degree of the product? Explain your reason and provide examples to support your explanation.
4. Think of a condition under which the product of any two binomials is a binomals. You can support your answer with the help of one of the below.
Is "2.1273685 ...." a rational or irrational number? Explain
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! Please help!
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1. Using one of the laws of exponents, prove that nay number raised to the power 0 is 1.
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one of the laws of exponents is that a^b * a^c = a^(b+c)
using this fact, you should be able to prove that a^0 = 1.
if you let c = 0, then this equation becomes:
a^b * a^0 = a^(b+0) = a^b
you have:
a^b * a^0 = a^b
if you divide both sides of this equation by a^b, you get:
a^0 = 1
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2. Using FOIL, simplify the expression (3x+2)(3x-2). Show that a particular factoring formula leads to the same answer.
FOIL is:
First: 3x*3x = 9x^2
Outer: 3x*-2 = -6x
Inner: 3x*2 = 6x
Last: 2*-2 = -4
putting them all together you have:
9x^2 - 6x + 6x - 4
combining like terms you get:
9x^2 -4
to find the roots you set this equation equal to 0.
you get:
9x^2 - 4 = 0
factoring formula is x = (-b +/- square root of (b^2 - 4ac))/2a
that's the factoring formula for a quadratic equation.
since the standard form of a quadratic equation is ax^2 + bx + c, then:
a = 9
b = 0
c = -4
formula becomes:
(-0 +/- square root of (0 - (4*-4*9)) / 18
which becomes:
+/- square root of (144) / 18
which becomes:
+/- 12/18
which becomes:
x = +/- 2/3
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this means that:
x = 2/3
or x = -2/3
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removing the denominators, we get:
3x = 2
or
3x = -2
subtracting 2 from both sides of this equation, we get:
(3x-2) = 0
(3x+2) = 0
since these are the original factors, we are good.
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3. If a fourths-sdegree polynomial is multiplied by third-degree polnomial, what degree of the product? Explain your reason and provide examples to support your explanation.
the product of a 4th degree polynomial and a third degree polynomial yields a 7th degree polynomial since the highest level of exponents that can be attained by the multiplication.
this is because:
x^a * x^b = x^(a+b)
example:
(x^4 + x^3 + x^2 + x) * (x^3 + x^2)
this equals:
x^7 + x^6 + x^5 + x^4
+
x^6 + x^5 + x^4 + x^3
since each term of one of the factors needs to be multiplied by each term of the other factor. This is an extension of the FOIL concept for quadratic equations which is really a sub form of this rule.
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4. Think of a condition under which the product of any two binomials is a binomals. You can support your answer with the help of one of the below.
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since a binomial is a polynomial with 2 terms, then any multiplication of 2 binomials where factors cancel out would yield another binomial.
take:
(x^2-2) * (x^2+2)
this yields:
x^4 - 2x^2 + 2x^2 - 4
which becomes:
x^4 - 4
which is a binomial because it has 2 terms.
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Is "2.1273685 ...." a rational or irrational number? Explain
a rational number is a number that can be expressed as the ratio of 2 numbers.
if this can, then it is a rational number.
if it can't, then it is not.
the fact that the decimal portion doesn't have a repeating pattern indicates that it is not.
example:
1/3 = .3333333
this has a repeating pattern and is clearly the result of the division of two integers (ratio of 2 integers is what the definition of a rational number is).
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best i can do now.
hope i answered your questions.
if you want more than you ever wanted to know about this, check out this website.
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http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm
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