MHB What Axioms Justify the Simplification of Polynomial Expressions?

[paulmdrdo1]

in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.

 

[topsquark]

in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.

You first have to prove: (a + b) + c = a + (b + c) = a + b + c. (I'm assuming the final form is meant to suggest addition of the terms in any order.)

Then for problem 1 use the above result to remove the parenthesis, use commutivity of addition to rearrange the terms, then use the distributive property to factor.

-Dan

 

[paulmdrdo1]

why did you use associativity of addition?

 

[paulmdrdo1]

i still don't understand what the question means.

 

[topsquark]

i still don't understand what the question means.

I'm assuming that if the addition is associative and commutative then we can show
(a + b) + c = (a + c) + b = (b + c) + a ... = a + b + c because we can show that order doesn't matter. So we simply call it a + b + c.

The problem is asking you to use this to remove the parenthesis in the following:
(x^2 + 2x + 5) + (x^2 + 3x + 1) = x^2 + 2x + 5 + x^2 + 3x + 1

To get to the final form you can use commutivity to rearrange the terms, then use the distributive property to factor them to the final form.

-Dan

 

[Evgeny.Makarov]

tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.

The answer to this question should be a list of axioms. The axioms in question are used in a proof of the equality above. Roughly speaking, a proof in this case is a chain of expressions $E_1=E_2=\dots=E_n$ where each $E_i$ has some subexpression $e$, $E_{i+1}$ is obtained from $E_i$ by replacing $e$ with $e'$ and $e=e'$ or $e'=e$ is an instance of an axiom of real numbers. For example, a proof may start with \[(x^2 + 2x + 5) + (x^2 + 3x + 1)=(1\cdot x^2 + 2x + 5) + (x^2 + 3x + 1)\]Here $E_1$ is $(x^2 + 2x + 5) + (x^2 + 3x + 1)$, $e$ is $x^2$ and $e'$ is $1\cdot x^2$. The axiom used here is $1\cdot x=x$ for all $x$, and $1\cdot x^2=x^2$ is its instance.

So you need to list all axioms that are used in the chain of equalities \[(x^2+2x+5)+(x^2+3x+1)=\dots=(1+1)x^2+(2+3)x+ (5+1)\]