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Greatest Common Factor and Factor by Grouping

Find the greatest common factor of two or more expressions

Find the Greatest Common Factor of Two or More Expressions

Earlier 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.

We summarize the steps we use to find the greatest common factor.

The next example will show us the steps to find the greatest common factor of three expressions.

Example

Try it.

Find the greatest common factor of \(21{x}^{3},9{x}^{2},15x.\)

Solution

Factor each coefficient into primes and write the
variables with exponents in expanded form.
Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
The GCF of \(21{x}^{3}\), \(9{x}^{2}\) and \(15x\) is \(3x\).

Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as \(2\cdot 6\) or \(3\cdot 4.\) In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as \(3{x}^{2}+15x,\) 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.”

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!

How to Use the Distributive Property to factor a polynomial

Try it.

Factor: \(8{m}^{3}-12{m}^{2}n+20m{n}^{2}.\)

Solution
Example

Try it.

Factor: \(5{x}^{3}-25{x}^{2}.\)

Solution

Find the GCF of \(5{x}^{3}\) and \(25{x}^{2}.\)
Rewrite each term.
Factor the GCF.
Check:

\(\ \begin{array}{l}5{x}^{2}(x-5) \\ 5{x}^{2}\cdot x-5{x}^{2}\cdot 5 \\ 5{x}^{3}-25{x}^{2}✓\end{array}\)

Example

Try it.

Factor: \(8{x}^{3}y-10{x}^{2}{y}^{2}+12x{y}^{3}.\)

Solution

The GCF of \(8{x}^{3}y,-10{x}^{2}{y}^{2},\ \text{and}\ 12x{y}^{3}\)
is \(2xy.\)
     
Rewrite each term using the GCF, \(2xy.\)      
Factor the GCF.     
Check:

\(\ \begin{array}{l}2xy(4{x}^{2}-5xy+6{y}^{2}) \\ 2xy\cdot 4{x}^{2}-2xy\cdot 5xy+2xy\cdot 6{y}^{2} \\ 8{x}^{3}y-10{x}^{2}{y}^{2}+12x{y}^{3}✓\end{array}\)

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Factor by Grouping

Sometimes 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.

How to Factor a Polynomial by Grouping

Try it.

Factor by grouping: \(xy+3y+2x+6.\)

Solution
Example

Try it.

Factor by grouping: ⓐ \({x}^{2}+3x-2x-6\) ⓑ \(6{x}^{2}-3x-4x+2.\)

Solution

ⓐ

There is no GCF in all four terms.\({x}^{2}+3x-2x-6\)
Separate into two parts.\({x}^{2}+3x\ -2x-6\)
Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.\(x(x+3)-2(x+3)\)
Factor out the common factor.\((x+3)(x-2)\)
Check on your own by multiplying.
ⓑ
There is no GCF in all four terms.\(6{x}^{2}-3x-4x+2\)
Separate into two parts.\(6{x}^{2}-3x\ -4x+2\)
Factor the GCF from both parts.\(3x(2x-1)-2(2x-1)\)
Factor out the common factor.\((2x-1)(3x-2)\)
Check on your own by multiplying.

Key Concepts

  • How to find the greatest common factor (GCF) of two expressions.
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors.
  • Distributive Property: If a, b, and c are real numbers, then
    \[a(b+c)=ab+ac\ \text{and}\ ab+ac=a(b+c)\]
    The form on the left is used to multiply. The form on the right is used to factor.
  • How to factor the greatest common factor from a polynomial.
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the “reverse” Distributive Property to factor the expression.
    4. Check by multiplying the factors.
  • Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.
    \[\begin{array}{llll}\text{Noun:} & & & \text{7 is a}\ \text{factor}\ \text{of 14} \\ \text{Verb:} & & & \text{factor}\ \text{3 from}\ 3a+3\end{array}\]
  • How to factor by grouping.
    1. Group terms with common factors.
    2. Factor out the common factor in each group.
    3. Factor the common factor from the expression.
    4. Check by multiplying the factors.

Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Try it.

\(10{p}^{3}q,12p{q}^{2}\)

Solution

\(2pq\)

Try it.

\(8{a}^{2}{b}^{3},10a{b}^{2}\)

Try it.

\(12{m}^{2}{n}^{3},30{m}^{5}{n}^{3}\)

Solution

\(6{m}^{2}{n}^{3}\)

Try it.

\(28{x}^{2}{y}^{4},42{x}^{4}{y}^{4}\)

Try it.

\(10{a}^{3},12{a}^{2},14a\)

Solution

\(2a\)

Try it.

\(20{y}^{3},28{y}^{2},40y\)

Try it.

\(35{x}^{3}{y}^{2},10{x}^{4}y,5{x}^{5}{y}^{3}\)

Solution

\(5{x}^{3}y\)

Try it.

\(27{p}^{2}{q}^{3},45{p}^{3}{q}^{4},9{p}^{4}{q}^{3}\)

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

Try it.

\(6m+9\)

Solution

\(3(2m+3)\)

Try it.

\(14p+35\)

Try it.

\(9n-63\)

Solution

\(9(n-7)\)

Try it.

\(45b-18\)

Try it.

\(3{x}^{2}+6x-9\)

Solution

\(3({x}^{2}+2x-3)\)

Try it.

\(4{y}^{2}+8y-4\)

Try it.

\(8{p}^{2}+4p+2\)

Solution

\(2(4{p}^{2}+2p+1)\)

Try it.

\(10{q}^{2}+14q+20\)

Try it.

\(8{y}^{3}+16{y}^{2}\)

Solution

\(8{y}^{2}(y+2)\)

Try it.

\(12{x}^{3}-10x\)

Try it.

\(5{x}^{3}-15{x}^{2}+20x\)

Solution

\(5x({x}^{2}-3x+4)\)

Try it.

\(8{m}^{2}-40m+16\)

Try it.

\(24{x}^{3}-12{x}^{2}+15x\)

Solution

\(3x(8{x}^{2}-4x+5)\)

Try it.

\(24{y}^{3}-18{y}^{2}-30y\)

Try it.

\(12x{y}^{2}+18{x}^{2}{y}^{2}-30{y}^{3}\)

Solution

\(6{y}^{2}(2x+3{x}^{2}-5y)\)

Try it.

\(21p{q}^{2}+35{p}^{2}{q}^{2}-28{q}^{3}\)

Try it.

\(20{x}^{3}y-4{x}^{2}{y}^{2}+12x{y}^{3}\)

Solution

\(4xy(5{x}^{2}-xy+3{y}^{2})\)

Try it.

\(24{a}^{3}b+6{a}^{2}{b}^{2}-18a{b}^{3}\)

Try it.

\(-2x-4\)

Solution

\(-2(x+2)\)

Try it.

\(-3b+12\)

Try it.

\(-2{x}^{3}+18{x}^{2}-8x\)

Solution

\(-2x({x}^{2}-9x+4)\)

Try it.

\(-5{y}^{3}+35{y}^{2}-15y\)

Try it.

\(-4{p}^{3}q-12{p}^{2}{q}^{2}+16p{q}^{2}\)

Solution

\(-4pq({p}^{2}+3pq-4q)\)

Try it.

\(-6{a}^{3}b-12{a}^{2}{b}^{2}+18a{b}^{2}\)

Try it.

\(5x(x+1)+3(x+1)\)

Solution

\((x+1)(5x+3)\)

Try it.

\(2x(x-1)+\ 9(x-1)\)

Try it.

\(3b(b-2)-13(b-2)\)

Solution

\((b-2)(3b-13)\)

Try it.

\(6m(m-5)-7(m-5)\)

Factor by Grouping

In the following exercises, factor by grouping.

Try it.

\(ab+5a+3b+15\)

Solution

\((b+5)(a+3)\)

Try it.

\(cd+6c+4d+24\)

Try it.

\(8{y}^{2}+y+40y+5\)

Solution

\((y+5)(8y+1)\)

Try it.

\(6{y}^{2}+7y+24y+28\)

Try it.

\(uv-9u+2v-18\)

Solution

\((u+2)(v-9)\)

Try it.

\(pq-10p+8q-80\)

Try it.

\({u}^{2}-u+6u-6\)

Solution

\((u-1)(u+6)\)

Try it.

\({x}^{2}-x+4x-4\)

Try it.

\(9{p}^{2}+12p-15p-20\)

Solution

\((3p-5)(3p+4)\)

Try it.

\(16{q}^{2}+20q-28q-35\)

Try it.

\(mn-6m-4n+24\)

Solution

\((n-6)(m-4)\)

Try it.

\({r}^{2}-3r-r+3\)

Try it.

\(2{x}^{2}-14x-5x+35\)

Solution

\((x-7)(2x-5)\)

Try it.

\(4{x}^{2}-36x-3x+27\)

Mixed Practice

In the following exercises, factor.

Try it.

\(-18x{y}^{2}-27{x}^{2}y\)

Solution

\(-9xy(2y+3x)\)

Try it.

\(-4{x}^{3}{y}^{5}-{x}^{2}{y}^{3}+12x{y}^{4}\)

Try it.

\(3{x}^{3}-7{x}^{2}+6x-14\)

Solution

\(({x}^{2}+2)(3x-7)\)

Try it.

\({x}^{3}+{x}^{2}+x+1\)

Try it.

\({x}^{2}+xy+5x+5y\)

Solution

\((x+y)(x+5)\)

Try it.

\(5{x}^{3}-3{x}^{2}+5x-3\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing 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.

First we’ll find the GCF of two numbers.

How to Find the Greatest Common Factor of Two or More Expressions

Try it.

Find the GCF of 54 and 36.

Solution

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

\[\begin{array}{l}54=18\cdot 3 \\ 36=18\cdot 2\end{array}\]

We summarize the steps we use to find the GCF below.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example

Try it.

Find the greatest common factor of \(27{x}^{3}\ \text{and}\ 18{x}^{4}\).

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
The GCF of \(27{x}^{3}\) and \(18{x}^{4}\) is \(9{x}^{3}.\)

Example

Try it.

Find the GCF of \(4{x}^{2}y,6x{y}^{3}\).

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
The GCF of \(4{x}^{2}y\) and \(6x{y}^{3}\) is \(2\text{xy}\).

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as \(2\cdot 6\ \text{or}\ 3\cdot 4),\) in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

\[\begin{array}{llll}2(x+7) & & & \text{factors} \\ 2\cdot x+2\cdot 7 & & & \\ 2x+14 & & & \text{product}\end{array}\]

Now we will start with a product, like \(2x+14\), and end with its factors, \(2(x+7)\). 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.”

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!

How to Factor the Greatest Common Factor from a Polynomial

Try it.

Factor: \(4x+12\).

Solution
Example

Try it.

Factor: \(5a+5\).

Solution

Find the GCF of 5a and 5.




Rewrite each term as a product using the GCF.
Use the Distributive Property "in reverse" to factor the GCF.
Check by mulitplying the factors to get the orginal polynomial.
\(5(a+1)\)
\(5⋅a+5⋅1\)
\(5a+5✓\)

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example

Try it.

Factor: \(12x-60\).

Solution

Find the GCF of 12x and 60.




Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying the factors.
\(12(x-5)\)
\(12⋅x-12⋅5\)
\(12x-60✓\)

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating 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.)

How to Factor by Grouping

Try it.

Factor: \(xy+3y+2x+6\).

Solution
Example

Try it.

Factor: \({x}^{2}+3x-2x-6\).

Solution

\(\begin{array}{llll}\text{There is no GCF in all four terms.} & & & \ {x}^{2}+3x\ -2x-6 \\ \text{Separate into two parts.} & & & \ \underset{⎵}{{x}^{2}+3x}\ \underset{⎵}{-2x-6} \\ \\ \\ \begin{array}{l}\text{Factor the GCF from both parts. Be careful} \\ \text{with the signs when factoring the GCF from} \\ \text{the last two terms.}\end{array} & & & \ \begin{array}{l}x(x+3)-2(x+3) \\ (x+3)(x-2)\end{array} \\ \\ \\ \text{Check on your own by multiplying.} & & & \end{array}\)

Key Concepts

  • Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors as in .
  • Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the ‘reverse’ Distributive Property to factor the expression.
    4. Check by multiplying the factors as in .
  • Factor by Grouping: To factor a polynomial with 4 four or more terms
    1. Group terms with common factors.
    2. Factor out the common factor in each group.
    3. Factor the common factor from the expression.
    4. Check by multiplying the factors as in .

Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

Try it.

8, 18

Solution

2

Try it.

24, 40

Try it.

72, 162

Solution

18

Try it.

150, 275

Try it.

10a, 50

Solution

10

Try it.

5b, 30

Try it.

\(3x,10{x}^{2}\)

Solution

\(x\)

Try it.

\(21{b}^{2},14b\)

Try it.

\(8{w}^{2},24{w}^{3}\)

Solution

\(8{w}^{2}\)

Try it.

\(30{x}^{2},18{x}^{3}\)

Try it.

\(10{p}^{3}q,12p{q}^{2}\)

Solution

\(2pq\)

Try it.

\(8{a}^{2}{b}^{3},10a{b}^{2}\)

Try it.

\(12{m}^{2}{n}^{3},30{m}^{5}{n}^{3}\)

Solution

\(6{m}^{2}{n}^{3}\)

Try it.

\(28{x}^{2}{y}^{4},42{x}^{4}{y}^{4}\)

Try it.

\(10{a}^{3},12{a}^{2},14a\)

Solution

\(2a\)

Try it.

\(20{y}^{3},28{y}^{2},40y\)

Try it.

\(35{x}^{3},10{x}^{4},5{x}^{5}\)

Solution

\(5{x}^{3}\)

Try it.

\(27{p}^{2},45{p}^{3},9{p}^{4}\)

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

Try it.

\(4x+20\)

Solution

\(4(x+5)\)

Try it.

\(8y+16\)

Try it.

\(6m+9\)

Solution

\(3(2m+3)\)

Try it.

\(14p+35\)

Try it.

\(9q+9\)

Solution

\(9(q+1)\)

Try it.

\(7r+7\)

Try it.

\(8m-8\)

Solution

\(8(m-1)\)

Try it.

\(4n-4\)

Try it.

\(9n-63\)

Solution

\(9(n-7)\)

Try it.

\(45b-18\)

Try it.

\(3{x}^{2}+6x-9\)

Solution

\(3({x}^{2}+2x-3)\)

Try it.

\(4{y}^{2}+8y-4\)

Try it.

\(8{p}^{2}+4p+2\)

Solution

\(2(4{p}^{2}+2p+1)\)

Try it.

\(10{q}^{2}+14q+20\)

Try it.

\(8{y}^{3}+16{y}^{2}\)

Solution

\(8{y}^{2}(y+2)\)

Try it.

\(12{x}^{3}-10x\)

Try it.

\(5{x}^{3}-15{x}^{2}+20x\)

Solution

\(5x({x}^{2}-3x+4)\)

Try it.

\(8{m}^{2}-40m+16\)

Try it.

\(12x{y}^{2}+18{x}^{2}{y}^{2}-30{y}^{3}\)

Solution

\(6{y}^{2}(2x+3{x}^{2}-5y)\)

Try it.

\(21p{q}^{2}+35{p}^{2}{q}^{2}-28{q}^{3}\)

Try it.

\(-2x-4\)

Solution

\(-2(x+2)\)

Try it.

\(-3b+12\)

Try it.

\(5x(x+1)+3(x+1)\)

Solution

\((x+1)(5x+3)\)

Try it.

\(2x(x-1)+9(x-1)\)

Try it.

\(3b(b-2)-13(b-2)\)

Solution

\((b-2)(3b-13)\)

Try it.

\(6m(m-5)-7(m-5)\)

Factor by Grouping

In the following exercises, factor by grouping.

Try it.

\(xy+2y+3x+6\)

Solution

\((y+3)(x+2)\)

Try it.

\(mn+4n+6m+24\)

Try it.

\(uv-9u+2v-18\)

Solution

\((u+2)(v-9)\)

Try it.

\(pq-10p+8q-80\)

Try it.

\({b}^{2}+5b-4b-20\)

Solution

\((b-4)(b+5)\)

Try it.

\({m}^{2}+6m-12m-72\)

Try it.

\({p}^{2}+4p-9p-36\)

Solution

\((p-9)(p+4)\)

Try it.

\({x}^{2}+5x-3x-15\)

Mixed Practice

In the following exercises, factor.

Try it.

\(-20x-10\)

Solution

\(-10(2x+1)\)

Try it.

\(5{x}^{3}-{x}^{2}+x\)

Try it.

\(3{x}^{3}-7{x}^{2}+6x-14\)

Solution

\(({x}^{2}+2)(3x-7)\)

Try it.

\({x}^{3}+{x}^{2}-x-1\)

Try it.

\({x}^{2}+xy+5x+5y\)

Solution

\((x+y)(x+5)\)

Try it.

\(5{x}^{3}+3{x}^{2}-5x-3\)

Try it.

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression \({w}^{2}-6w\), where \(w=\) width. Factor the greatest common factor from the polynomial.

Solution

\(w(w-6)\)

Try it.

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression \(-16{t}^{2}+80t+4\). Factor the greatest common factor from the polynomial.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Factor 56 into primes.
    If you missed this problem, review .

    Giải đáp

    \(2\cdot 2\cdot 2\cdot 7\)

  2. Find the least common multiple (LCM) of 18 and 24.
    If you missed this problem, review .

    Giải đáp

    72

  3. Multiply: \(-3a(7a+8b).\)
    If you missed this problem, review .

    Giải đáp

    \(-21{a}^{2}-24ab\)

  4. Find the greatest common factor of \(21{x}^{3},9{x}^{2},15x.\)

    Giải đáp

    Factor each coefficient into primes and write the
    variables with exponents in expanded form.
    Circle the common factors in each column.
    Bring down the common factors.
    Multiply the factors.
    The GCF of \(21{x}^{3}\), \(9{x}^{2}\) and \(15x\) is \(3x\).

  5. Find the greatest common factor: \(25{m}^{4},35{m}^{3},20{m}^{2}.\)

    Giải đáp

    \(5{m}^{2}\)

  6. Find the greatest common factor: \(14{x}^{3},70{x}^{2},105x.\)

    Giải đáp

    \(7x\)

  7. Factor: \(8{m}^{3}-12{m}^{2}n+20m{n}^{2}.\)

  8. Factor: \(9x{y}^{2}+6{x}^{2}{y}^{2}+21{y}^{3}.\)

    Giải đáp

    \(3{y}^{2}(3x+2{x}^{2}+7y)\)

  9. Factor: \(3{p}^{3}-6{p}^{2}q+9p{q}^{3}.\)

    Giải đáp

    \(3p({p}^{2}-2pq+3{q}^{2})\)

  10. Factor: \(5{x}^{3}-25{x}^{2}.\)

    Giải đáp

    Find the GCF of \(5{x}^{3}\) and \(25{x}^{2}.\)
    Rewrite each term.
    Factor the GCF.
    Check:

    \(\ \begin{array}{l}5{x}^{2}(x-5) \\ 5{x}^{2}\cdot x-5{x}^{2}\cdot 5 \\ 5{x}^{3}-25{x}^{2}✓\end{array}\)

  11. Factor: \(2{x}^{3}+12{x}^{2}.\)

    Giải đáp

    \(2{x}^{2}(x+6)\)

  12. Factor: \(6{y}^{3}-15{y}^{2}.\)

    Giải đáp

    \(3{y}^{2}(2y-5)\)

  13. Factor: \(8{x}^{3}y-10{x}^{2}{y}^{2}+12x{y}^{3}.\)

    Giải đáp

    The GCF of \(8{x}^{3}y,-10{x}^{2}{y}^{2},\ \text{and}\ 12x{y}^{3}\)
    is \(2xy.\)
         
    Rewrite each term using the GCF, \(2xy.\)      
    Factor the GCF.     
    Check:

    \(\ \begin{array}{l}2xy(4{x}^{2}-5xy+6{y}^{2}) \\ 2xy\cdot 4{x}^{2}-2xy\cdot 5xy+2xy\cdot 6{y}^{2} \\ 8{x}^{3}y-10{x}^{2}{y}^{2}+12x{y}^{3}✓\end{array}\)

  14. Factor: \(15{x}^{3}y-3{x}^{2}{y}^{2}+6x{y}^{3}.\)

    Giải đáp

    \(3xy(5{x}^{2}-xy+2{y}^{2})\)

  15. Factor: \(8{a}^{3}b+2{a}^{2}{b}^{2}-6a{b}^{3}.\)

    Giải đáp

    \(2ab(4{a}^{2}+ab-3{b}^{2})\)

  16. Factor: \(-4{a}^{3}+36{a}^{2}-8a.\)

    Giải đáp

    The leading coefficient is negative, so the GCF will be negative.

    Rewrite each term using the GCF, \(-4a.\)
    Factor the GCF.
    Check:

    \(\ \begin{array}{l}-4a({a}^{2}-9a+2) \\ -4a\cdot {a}^{2}-(-4a)\cdot 9a+(-4a)\cdot 2 \\ -4{a}^{3}+36{a}^{2}-8a✓\end{array}\)

  17. Factor: \(-4{b}^{3}+16{b}^{2}-8b.\)

    Giải đáp

    \(-4b({b}^{2}-4b+2)\)

  18. Factor: \(-7{a}^{3}+21{a}^{2}-14a.\)

    Giải đáp

    \(-7a({a}^{2}-3a+2)\)

  19. Factor: \(3y(y+7)-4(y+7).\)

    Giải đáp

    The GCF is the binomial \(y+7.\)

    Factor the GCF, \((y+7).\)
    Check on your own by multiplying.   

  20. Factor: \(4m(m+3)-7(m+3).\)

    Giải đáp

    \((m+3)(4m-7)\)

  21. Factor: \(8n(n-4)+5(n-4).\)

    Giải đáp

    \((n-4)(8n+5)\)

  22. Factor by grouping: \(xy+3y+2x+6.\)

  23. Factor by grouping: \(xy+8y+3x+24.\)

    Giải đáp

    \((x+8)(y+3)\)

  24. Factor by grouping: \(ab+7b+8a+56.\)

    Giải đáp

    \((a+7)(b+8)\)

  25. Factor by grouping: ⓐ \({x}^{2}+3x-2x-6\) ⓑ \(6{x}^{2}-3x-4x+2.\)

    Giải đáp

    ⓐ

    There is no GCF in all four terms.\({x}^{2}+3x-2x-6\)
    Separate into two parts.\({x}^{2}+3x\ -2x-6\)
    Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.\(x(x+3)-2(x+3)\)
    Factor out the common factor.\((x+3)(x-2)\)
    Check on your own by multiplying.
    ⓑ
    There is no GCF in all four terms.\(6{x}^{2}-3x-4x+2\)
    Separate into two parts.\(6{x}^{2}-3x\ -4x+2\)
    Factor the GCF from both parts.\(3x(2x-1)-2(2x-1)\)
    Factor out the common factor.\((2x-1)(3x-2)\)
    Check on your own by multiplying.

  26. Factor by grouping: ⓐ \({x}^{2}+2x-5x-10\) ⓑ \(20{x}^{2}-16x-15x+12.\)

    Giải đáp

    ⓐ \((x-5)(x+2)\)
    ⓑ \((5x-4)(4x-3)\)

  27. Factor by grouping: ⓐ \({y}^{2}+4y-7y-28\) ⓑ \(42{m}^{2}-18m-35m+15.\)

    Giải đáp

    ⓐ \((y+4)(y-7)\)
    ⓑ \((7m-3)(6m-5)\)

  28. \(10{p}^{3}q,12p{q}^{2}\)

    Giải đáp

    \(2pq\)

  29. \(8{a}^{2}{b}^{3},10a{b}^{2}\)

  30. \(12{m}^{2}{n}^{3},30{m}^{5}{n}^{3}\)

    Giải đáp

    \(6{m}^{2}{n}^{3}\)

  31. \(28{x}^{2}{y}^{4},42{x}^{4}{y}^{4}\)

  32. \(10{a}^{3},12{a}^{2},14a\)

    Giải đáp

    \(2a\)

  33. \(20{y}^{3},28{y}^{2},40y\)

  34. \(35{x}^{3}{y}^{2},10{x}^{4}y,5{x}^{5}{y}^{3}\)

    Giải đáp

    \(5{x}^{3}y\)

  35. \(27{p}^{2}{q}^{3},45{p}^{3}{q}^{4},9{p}^{4}{q}^{3}\)

  36. \(3{x}^{2}+6x-9\)

    Giải đáp

    \(3({x}^{2}+2x-3)\)

  37. \(4{y}^{2}+8y-4\)

  38. \(8{p}^{2}+4p+2\)

    Giải đáp

    \(2(4{p}^{2}+2p+1)\)

  39. \(10{q}^{2}+14q+20\)

  40. \(8{y}^{3}+16{y}^{2}\)

    Giải đáp

    \(8{y}^{2}(y+2)\)

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Greatest Common Factor and Factor by Grouping

  1. Find the greatest common factor of two or more expressions
  2. Factor the greatest common factor from a polynomial
  3. Factor by grouping
  4. Factor each coefficient into primes. Write all variables with exponents in expanded form.
  5. List all factors—matching common factors in a column. In each column, circle the common factors.
  6. Bring down the common factors that all expressions share.
  7. Multiply the factors.
  8. Find the GCF of all the terms of the polynomial.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Thử đi.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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