factorization

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, $4x(x+5) + 3y(x+5)$.
  • Factor out the binomial $(x+5)(4x+3y)$.
  • One way to factor polynomials is factoring by grouping.
  • Both groups share the same factor $(x+5)$, so the polynomial is factored as:

• Factoring General Quadratics

  • We can factor quadratic equations of the form $ax^2 + bx + c$ by first finding the factors of the constant $c$.  
  • This leads to the factored form:
  • First, we factor $a$, which has one pair of factors 3 and 2.
  • Then we factor the constant $c$, 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 $4 \cdot x$, 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 $3$.
  • This is the simplest kind of factoring.
  • Therefore it factors as $(x+5)(x-5)$.

• 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 $x^2-7x+12$.
  • 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 $x$-axis.
  • In general, we know from the remainder theorem that $a$ is a zero of $f(x)$ if and only if $x-a$ divides $f(x).$ Thus if we can factor $f(x)$ 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: $(x^2 + 2)$.

• Factoring a Difference of Squares

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

• Rules for Exponent Arithmetic

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