What is the differenceand sum of cubes?
Similar to squares, there is also the difference and sum of cubes that deals with the
factoring of polynomials. The difference of cubestakes the form a^3 b^3 , and can be
factored into (a b)(a^2 ab b^2 ). Thus, if an expression resembles (a^3 b^3 ), then (a
b) is a factor; use long division to find the remaining factor(s).
The sum of cubestakes the form a^3 b^3 , and can be factored into (a b)(a^2 ab
b^2 ). Thus, if an expression resembles (a^3 b^3 ), then (a b) is a factor. Again, use
long division to find the remaining factor(s).
How do you find the rootsof a polynomial?
Finding the root, also called a zero, of a polynomial is one way to solve for the equa-
tion. In other words, the root of an equation is simply a number (or numbers) that
solves the equation.
For example, for second-degree polynomials we can find the roots by completing
the square. Picking apart an equation is the best way to see this:
- 3x^2 4x 1 0
- (1/3)(3x^2 4x 1) (1/3)0 (making the coefficient of the x^2 term into a 1)
- x^2 (4/3)x 1/3 0
- (x^2 (4/3)x) 1/3 0 (group the xand x^2 terms together)
- (x^2 (4/3)x (2/3)^2 ) (2/3)^2 1/3 0 (determine the coefficient of the x
term, divide it by 2 and then square; add and subtract that term) 153
ALGEBRA
What is a perfect square?
T
here are many equations that can be factored into a perfect square. Any expres-
sion written in the form x^2 2 axa^2 is a perfect square—an expression writ-
ten as [something]^2. To determine if an expression is a perfect square, first see if
the constant term is a square number—in other words, can the square root of the
number be taken to get an integer for an answer. If so, determine if the square root
of the constant, multiplied by 2, gives the coefficient of the linear term (or the x
term). If it does, the original expression may be factored into a perfect square.
(Note: The above procedure only works when the coefficient of x^2 is 1.)For example, in the equation x^2 ^8 x16, the constant term (16) is already
a perfect square (the square root of 16 is 4). Since 2(4) 8, the original expres-
sion can be written as a perfect square. Because we know x^2 2 axa^2 is a per-
fect square, and equals (xa)^2 , by substituting the common factor 4 into the
equation, we find that x^2 8 x 16 (x4)^2.