By the end of this section, you will be able to: Be PreparedBefore you get started, take this readiness quiz.
Find the Greatest Common Factor of Two or More ExpressionsEarlier we multiplied factors together to get a product. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM. GREATEST COMMON FACTORThe greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. We summarize the steps we use to find the greatest common factor. FIND THE GREATEST COMMON FACTOR (GCF) OF TWO EXPRESSIONS.
The next example will show us the steps to find the greatest common factor of three expressions. Example \(\PageIndex{1}\)Find the greatest common factor of \(21x^3,\space 9x^2,\space 15x\). Answer
Example \(\PageIndex{2}\)Find the greatest common factor: \(25m^4,\space 35m^3,\space 20m^2.\) Answer\(5m^2\) Example \(\PageIndex{3}\)Find the greatest common factor: \(14x^3,\space 70x^2,\space 105x\). Answer\(7x\) Factor the Greatest Common Factor from a PolynomialIt is sometimes useful to represent a number as a product of factors, for example, 12 as \(2·6\) or \(3·4\). In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as \(3x^2+15x\), and end with its factors, \(3x(x+5)\). To do this we apply the Distributive Property “in reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.” DISTRIBUTIVE PROPERTYIf a, b, and c are real numbers, then \[a(b+c)=ab+ac \quad \text{and} \quad ab+ac=a(b+c)\nonumber\] The form on the left is used to multiply. The form on the right is used to factor. So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product! Example \(\PageIndex{4}\): How to Use the Distributive Property to factor a polynomialFactor: \(8m^3−12m^2n+20mn^2\). AnswerExample \(\PageIndex{5}\)Factor: \(9xy^2+6x^2y^2+21y^3\). Answer\(3y^2(3x+2x^2+7y)\) Example \(\PageIndex{6}\)Factor: \(3p^3−6p^2q+9pq^3\). Answer\(3p(p^2−2pq+3q^3)\) FACTOR THE GREATEST COMMON FACTOR FROM A POLYNOMIAL.
FACTOR AS A NOUN AND A VERBWe use “factor” as both a noun and a verb: \[\begin{array} {ll} \text{Noun:} &\hspace{50mm} 7 \text{ is a factor of }14 \\ \text{Verb:} &\hspace{50mm} \text{factor }3 \text{ from }3a+3\end{array}\nonumber\] Example \(\PageIndex{7}\)Factor: \(5x^3−25x^2\). Answer
Example \(\PageIndex{8}\)Factor: \(2x^3+12x^2\). Answer\(2x^2(x+6)\) Example \(\PageIndex{9}\)Factor: \(6y^3−15y^2\). Answer\(3y^2(2y−5)\) Example \(\PageIndex{10}\)Factor: \(8x^3y−10x^2y^2+12xy^3\). Answer
Example \(\PageIndex{11}\)Factor: \(15x^3y−3x^2y^2+6xy^3\). Answer\(3xy(5x^2−xy+2y^2)\)
Example \(\PageIndex{12}\)Factor: \(8a^3b+2a^2b^2−6ab^3\). Answer\(2ab(4a^2+ab−3b^2)\) When the leading coefficient is negative, we factor the negative out as part of the GCF. Example \(\PageIndex{13}\)Factor: \(−4a^3+36a^2−8a\). AnswerThe leading coefficient is negative, so the GCF will be negative.
Example \(\PageIndex{14}\)Factor: \(−4b^3+16b^2−8b\). Answer\(−4b(b^2−4b+2)\) Example \(\PageIndex{15}\)Factor: \(−7a^3+21a^2−14a\). Answer\(−7a(a^2−3a+2)\) So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial. Example \(\PageIndex{16}\)Factor: \(3y(y+7)−4(y+7)\). AnswerThe GCF is the binomial \(y+7\).
Example \(\PageIndex{17}\)Factor: \(4m(m+3)−7(m+3)\). Answer\((m+3)(4m−7)\) Example \(\PageIndex{18}\)Factor: \(8n(n−4)+5(n−4)\). Answer\((n−4)(8n+5)\) Factor by GroupingSometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime. Example \(\PageIndex{19}\): How to Factor a Polynomial by GroupingFactor by grouping: \(xy+3y+2x+6\). AnswerExample \(\PageIndex{20}\)Factor by grouping: \(xy+8y+3x+24\). Answer\((x+8)(y+3)\) Example \(\PageIndex{21}\)Factor by grouping: \(ab+7b+8a+56\). Answer\((a+7)(b+8)\) FACTOR BY GROUPING.
Example \(\PageIndex{22}\)Factor by grouping: ⓐ \(x^2+3x−2x−6\) ⓑ \(6x^2−3x−4x+2\). Answerⓐ ⓑ Example \(\PageIndex{23}\)Factor by grouping: ⓐ \(x^2+2x−5x−10\) ⓑ \(20x^2−16x−15x+12\). Answerⓐ \((x−5)(x+2)\) Example \(\PageIndex{24}\)Factor by grouping: ⓐ \(y^2+4y−7y−28\) ⓑ \(42m^2−18m−35m+15\). Answerⓐ \((y+4)(y−7)\) Key Concepts
GlossaryfactoringSplitting a product into factors is called factoring.greatest common factorThe greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. |