Product Rule of Exponents
Multiplication of the same constant or variable to various powers can be simplified by adding up their exponents:
53×54=57
This also holds true for distribution:
x2(x−2x2)=x3−2x4 because x=x1
4x2×9x7=36x9
x3(2x+3x2)=2x4+3x5 because x3=1×x3
(x−y)(x−y)3=(x−y)4
Power Rule of Exponents
Exponents distribute with multiplication:
(53)4=512
(bm)n=bm×n
Signs are to be interpreted as −1:
(−3x4)3=(−1×3×x4)3
=(−1)333x12
=(−1)×27×x12
The product rule is not used when constants are not the same.
Polynomials and Monomials
Polynomials are expressions linking multiple terms by addition and/or subtraction.
Degree
The degree of a term is the sum of all of its exponents. A polynomial's degree is the highest degree of the degrees of its terms.
Exceptions
An expression containing any of the following is not a polynomial:
- a fraction with a variable in the denominator
- any negative exponents
(7x2+3x−8)−(5x2+6)=7x2+3x−8−5x2−6
=2x2+3x−14
Distribution of Monomials
3y3(−7y4)=−21y7
−y(−2y+4)=2y−4y
2xy(7x2y3−3y)=14x3y4−6xy2
a4(3a−7)=3a5−7a4
Distribution of Binomials
All terms are to be multiplied together with the resulting terms linked by either addition or subtraction:
(x+3)(x+5)=(x×x)+(x×5)+(3×x)+(3×5)
=x2+8x+15
Special Products of Polynomials
-
(a−b)(a+b)=a2−b2 -> (2x2−5y3)(2x2+5y3)=4x4−25y6
-
(a+b)2=(a+b)(a+b)=a2+ab+ab+b2=a2+2ab+b2 -> (2y+4)2=4y2+16y+16
Dividing Polynomials by Monomials
Considering that 71+73=74, divisions of polynomials by monomials can be interpreted as the sum of two fractions with the same denominator:
x2x5+x3=x2x5+x2x3=x3+x
Long Division Algorithm for Polynomials
In some cases long division is the best way to handle the division of a polynomial by another polynomial:
x+12x3−x+1=x+1)2x3+0x2−x+1=2x2−2x+1
- Fill up missing powers
- Divide the first term on the right side (2x3) by the first term on the left side (x)
- Move result to the top of the line
- Distribute the same term with the terms on the left side, writing the result below the term on the right (while filling up missing powers)
- Subtract the polynomial below the line with the new one written beneath it
- Bring down the next term if it isn't already part of the result
- Repeat
- If the algorithm's last subtraction does not yield a 0, it has yielded a remainder. The remainder is brought up as numerator over the left polynomial (in this case x+1 as its denominator.
Negative Exponents
Negative exponents can be transformed into positive numbers by turning their base into the numerator of a fraction and turning that fraction into its reciprocal:
b−n -> 1b−n=bn1 -> 9−2=921=811
(x−y)−7=(x−y)71
3x−1=x13
5xy−1=y15x
Quotient Rule for Exponents
A number raised to a power divided by the same number is equal to that itself raised to the power of the exponents' difference.
bnbm=bm−n -> x2x3=x
x7x5=x−2=x21
7x3y728x6y2=4x3y51
b−n1=bn -> 5−41=54
b−na−m=ambn
(ba)−n=b−na−n=anbn
Complex terms with fractions
These can be interpreted as multiple individual fractions multiplied together:
x−3y−27x−3y−5=17×x−3x−3×y−2y−5=17×1×y31