factor
(noun)
Any of various objects multiplied together to form some whole.
(verb)
To express a mathematical quantity as a product of two or more like quantities.
(verb)
To find all the mathematical objects that divide a mathematical object evenly.
Examples of factor in the following topics:
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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.
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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:
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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
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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
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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 $(x+5)(x-5)$.
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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.
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Finding Zeros 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
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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.
- We begin problems of this type by factoring.
- Notice the factors in the denominators.
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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.
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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.