Examples of factoring in the following topics:
-
- 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.
-
- 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:
-
- We can factor quadratic equations of the form ax2+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
-
- 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⋅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
-
- 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).
-
- 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+12.
- We attempt to factor the quadratic.
-
- 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
-
- We start, as usual, by factoring.
- Similarly, the prime factors of 30 are 2, 3, and 5.
- This requires factoring algebraic expressions.
- We begin problems of this type by factoring.
- Notice the factors in the denominators.
-
- When a quadratic is a difference of squares, there is a helpful formula for factoring it.
- But x2=a2 can also be solved by rewriting the equation as x2−a2=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 4x2 is always greater than or equal to 0. However, the first equation 4x2−3=0 can be factored again as the difference of squares, if we consider 3 as the square of √3.
- Evaluate whether a quadratic equation is a difference of squares and factor it accordingly if it is.
-
- am means that you have m factors of a.
- If you multiply this quantity by an, i.e. by n additional factors of a, then you have am+n factors in total.
- In the same way that am⋅an=am+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.