How do you divide polynomials?
To divide a polynomial by a monomial, divide each term in the polynomial by the
monomial (to divide monomials, divide the coefficients and then subtract the expo-
nents) or:
(A B C)/M A/M B/M C/M
For example, to solve the equation10 x^5 /2x^3 (10/2) (x^5 ^3 ) 5 x^2What does factoring polynomialsmean?
When a polynomial is written as the product of two or more polynomial equations, the
polynomial has been factored. This allows a complicated polynomial to be broken up into
easier, lower-degree pieces; and it makes the equation easier to solve. One way to look at it
is that factoring a polynomial is the opposite process from multiplying polynomials.
One of the most basic ways to factor a polynomial is similar to factoring a number.
When a number is factored, the result will be the prime factors that multiply together
to give the number (for example, 6 2 3 , or 12 2 2 3; see “Math Basics” to
learn more about prime factors). With polynomials, this is often called “taking out a
common factor”: If every term in a polynomial expression has several factors, and if
every term has at least one factor that is the same, then that factor is called a common
factor. If this is the case, then the common factor can be removed from every term
and multiplied by the whole remaining expression.
For example, for the equation 2x^2 8 x,the first term has factors of 2 and x,while
the second term has factors of 2, 4, and x. The common factors are 2 and x,making 2x
the overall common factor. This makes the expression equal to 2x(x4). Thus, it is
easy to see that when a polynomial is factored, it results in simpler polynomials that
can be multiplied together to give the initial polynomial.What is differences of squares?
Differences of squares is another way to factor an expression into a form that is essen-
tially [something]^2 subtracted from [something]^2. Mathematically, an equation that
resembles (a^2 x^2 b^2 ) is a difference of squares and can be factored into (axb) (ax
b), with the factors being identical, except for the sign. This, in turn, equals
(ax)(ax) abxabxb^2. The two middle terms (abxand abx) further cancel
each other out, resulting in (a^2 x^2 b^2 ), which results in a way of factoring called the
difference of squares. For example, take the expression 16 s^2. From the above, we
know that the form (a^2 x^2 b^2 ) equals (axb)(axb). Thus, by “substitution,” and if
a1, x^2 16, and bs,we can say that 16 s^2 (4 s)(4 s), with the resulting
152 factoring of the equation using difference of squares.