factorization

(noun)

The process of creating a list of items that, when multiplied together, will produce a desired quantity or expression.

Related Terms

  • greatest common divisor
  • factor
  • prime number
  • prime factor
  • polynomial

(noun)

An expression listing items that, when multiplied together, will produce a desired quantity.

Related Terms

  • greatest common divisor
  • factor
  • prime number
  • prime factor
  • polynomial

Examples of factorization in the following topics:

  • Factors

    • This is a complete list of the factors of 24.
    • Therefore, 2 and 3 are prime factors of 6.
    • However, 6 is not a prime factor.
    • To factor larger numbers, it can be helpful to draw a factor tree.
    • This factor tree shows the factorization of 864.
  • Introduction to Factoring Polynomials

    • Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
    • Factor out the greatest common factor, 4x(x+5)+3y(x+5)4x(x+5) + 3y(x+5)4x(x+5)+3y(x+5).
    • Factor out the binomial (x+5)(4x+3y)(x+5)(4x+3y)(x+5)(4x+3y).
    • One way to factor polynomials is factoring by grouping.
    • Both groups share the same factor (x+5)(x+5)(x+5), so the polynomial is factored as:
  • Factoring General Quadratics

    • We can factor quadratic equations of the form ax2+bx+cax^2 + bx + cax​2​​+bx+c by first finding the factors of the constant ccc.  
    • This leads to the factored form:
    • First, we factor aaa, which has one pair of factors 3 and 2.
    • Then we factor the constant ccc, which has one pair of factors 2 and 4.
    • Using these factored sets, we assemble the final factored form of the quadratic
  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • It is important to be able to recognize such trinomials, so that they can the be factored as a perfect square.
    • If you are attempting to to factor a trinomial and realize that it is a perfect square, the factoring becomes much easier to do.
    • Since the middle term is twice 4⋅x4 \cdot x4⋅x, this must be a perfect square trinomial, and we can factor it as:
    • Evaluate whether a quadratic equation is a perfect square and factor it accordingly if it is
  • Finding Factors of Polynomials

    • When factoring, things are pulled apart.
    • There are four basic types of factoring.
    • The common factor is 333.
    • This is the simplest kind of factoring.
    • Therefore it factors as (x+5)(x−5)(x+5)(x-5)(x+5)(x−5).
  • Solving Quadratic Equations by Factoring

    • To factor an expression means to rewrite it so that it is the product of factors.
    • The reverse process is called factoring.
    • Factoring is useful to help solve an equation of the form:
    • Again, imagine you want to factor x2−7x+12x^2-7x+12x​2​​−7x+12.
    • We attempt to factor the quadratic.
  • Finding Zeroes of Factored Polynomials

    • The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the x axis.
    • The factored form of a polynomial can reveal where the function crosses the xxx-axis.
    • In general, we know from the remainder theorem that aaa is a zero of f(x)f(x)f(x) if and only if x−ax-ax−a divides f(x).f(x).f(x). Thus if we can factor f(x)f(x)f(x) in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.
    • This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly.
    • Use the factored form of a polynomial to find its zeros
  • Rational Algebraic Expressions

    • We start, as usual, by factoring.
    • Similarly, the prime factors of 30 are 2, 3, and 5.
    • This requires factoring algebraic expressions.
    • Notice the factors in the denominators.
    • The second fraction has one factor: (x2+2)(x^2 + 2)(x​2​​+2).
  • Factoring a Difference of Squares

    • When a quadratic is a difference of squares, there is a helpful formula for factoring it.
    • But x2=a2x^2 = a^2x​2​​=a​2​​ can also be solved by rewriting the equation as x2−a2=0x^2-a^2=0x​2​​−a​2​​=0 and factoring the difference of squares.
    • If you recognize the first term as the square of xxx and the term after the minus sign as the square of 444, you can then factor the expression as:
    • This latter equation has no solutions, since 4x24x^24x​2​​ is always greater than or equal to 0.0.0. However, the first equation 4x2−3=04x^2-3=04x​2​​−3=0 can be factored again as the difference of squares, if we consider 333 as the square of 3\sqrt3√​3​​​.
    • Evaluate whether a quadratic equation is a difference of squares and factor it accordingly if it is.
  • Rules for Exponent Arithmetic

    • ama^ma​m​​ means that you have mmm factors of aaa.
    • If you multiply this quantity by ana^na​n​​, i.e. by nnn additional factors of aaa, then you have am+na^{m+n}a​m+n​​ factors in total.
    • In the same way that am⋅an=am+n{ a }^{ m }\cdot { a }^{ n }={ a }^{ m+n }a​m​​⋅a​n​​=a​m+n​​ because you are adding on factors of aaa, dividing removes factors of aaa.
    • If you have nnn factors of aaa in the denominator, then you can cross out nnn factors from the numerator.
    • If there were mmm factors in the numerator, now you have (m−n)(m-n)(m−n) factors in the numerator.
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