Principle A.2.13. Closure of Addition and Multiplication on the Set of Real Numbers.
If \(a\) and \(b\) are real numbers, then \(a+b\) and \(a\cdot b\) are real numbers.
Subsection A.2 Field Properties of the Real Numbers
We summarize the principles that define operations on the set of real numbers. Methods of performing calculations, simplifying algebraic expressions, and solving equations follow from these principles.
Principle A.2.14. Commutative Laws.
If \(a\) and \(b\) are real numbers, then \(a+b=b+a\) and \(a\cdot b=b\cdot a\text{.}\)
Principle A.2.15. Associative Laws.
If \(a\text{,}\) \(b\text{,}\) and \(c\) are real numbers, then \((a+b)+c=a+(b+c)\) and \((a\cdot b)\cdot c=a\cdot (b\cdot c)\text{.}\)
Principle A.2.16. The Distributive Law.
If \(a\text{,}\) \(b\text{,}\) and \(c\) are real numbers, then \(c\cdot(a+b)=ca+cb\) and \((a+b)\cdot c=ac+bc\text{.}\)
Note that the distributive law also allows us to add fractions with a common denominator;
\begin{equation*}
\frac{x}{d}+\frac{y}{d}=\frac{1}{d}\cdot x+\frac{1}{d}\cdot y
=\frac{1}{d}(x+y)=\frac{x+y}{d}.
\end{equation*}
In short, \(\frac{x}{d}+\frac{y}{d}=\frac{x+y}{d}.\)
The Distributive Law, together with the other field properties, allows us to multiply polynomial.
Example A.2.17. Multiplying a Polynomial.
Multiply \((2x+1)(x^2-3x+5)\text{.}\)
Answer.
\(2x^3-5x^2+7x+5\)
Solution.
First we use the distributive law, multiplying each term in the second factor by \(2x+1\text{:}\) \((2x+1)(x^2-3x+5)=(2x+1)(x^2)+(2x+1)(-3x)+(2x+1)(5)\text{.}\)
An obtional step, but one that makes the distributive law more evident, is to us the commutative law of multiplication to reverse the order of the factors in each term: \((2x+1)(x^2)+(2x+1)(-3x)+(2x+1)(5)=x^2(2x+1)+(-3x)(2x+1)+5(2x+1).\)
We can then use the distributive law on each term to get \(x^2(2x+1)+(-3x)(2x+1)+5(2x+1)=x^2(2x)+x^2(1)-3x(2x)-3x(1)+5(2x)+5(1)
2x^3+x^2-6x^2-3x+10x+5 \)
Finally, we use the distributive law to combine like terms: \(2x^3+x^2-6x^2-3x+10x+5 =2x^3+(1-6)x^2+(10-3)x+5
=2x^3-5x^2+7x+5.\)
Although the distributive law makes the process work, we typically speed the process up by multiplying each term in the first factor by each term in the second factor: \((2x+1)(x^2-3x+5)=2x(x^2)+2x(-3x)+2x(5)+1(x^2)+1(-3x)+1(5)
=2x^3-6x^2+10x+x^2-3x+5
=2x^3-5x^2+7x+5.\)
As demonstrated in the solution of Example A.2.17, the product of two polynomials can be found by multiplying each term in the first polynomial by each term in the second polynomial and then adding these products. If the first polynomial has \(m\) terms and the second polynomial has \(n\) terms, then there will be \(mn\) terms before we combine like terms.
Principle A.2.18. The Identity Properties.
If \(a\) is an real number, then \(a+0=a\text{,}\) \(0+a=a\text{,}\) \(a\cdot 1=a\text{,}\) and \(1\cdot a=a.\) We call 0 the additive identity and 1 the multiplicative identity.
Principle A.2.19. The Inverse Properties.
If \(a\) is a real number, then the number \(-a\) exists and has the property that \(a+(-a)=0\text{.}\) If \(a\) is an nonzero real number, then the number \(\frac{1}{a}\) exists and has the property that \(a\cdot\frac{1}{a}=1\text{.}\) The number \(-a\) is said to be the additive inverse of \(a\) and the number \(\frac{1}{a}\) is said to be the multiplicative inverse of \(a\text{.}\)
