General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example

Try it.

Factor completely: \(7{x}^{3}-21{x}^{2}-70x.\)

Solution

	\(7{x}^{3}-21{x}^{2}-70x\)
Is there a GCF? Yes, \(7x\).
Factor out the GCF.	\(7x({x}^{2}-3x-10)\)
In the parentheses, is it a binomial, trinomial, or are there more terms?
Trinomial with leading coefficient 1.
“Undo” FOIL.	\(7x(x\ )(x\ )\)
	\(7x(x+2)(x-5)\)
Is the expression factored completely? Yes.
Neither binomial can be factored.
Check your answer.
Multiply.
\(7x(x+2)(x-5)\)
\(7x({x}^{2}-5x+2x-10)\)
\(7x({x}^{2}-3x-10)\)
\(7{x}^{3}-21{x}^{2}-70x✓\)

Be careful when you are asked to factor a binomial as there are several options!

Example

Try it.

Factor completely: \(24{y}^{2}-150.\)

Solution

	\(24{y}^{2}-150\)
Is there a GCF? Yes, 6.
Factor out the GCF.	\(6(4{y}^{2}-25)\)
In the parentheses, is it a binomial, trinomial or are there more than three terms? Binomial.
Is it a sum? No.
Is it a difference? Of squares or cubes? Yes, squares.	\(6({(2y)}^{2}-{(5)}^{2})\)
Write as a product of conjugates.	\(6(2y-5)(2y+5)\)
\(\ \text{Is the expression factored completely?}\)
\(\ \text{Neither binomial can be factored.}\)
Check:
\(\ \text{Multiply.}\)
\(6(2y-5)(2y+5)\)
\(6(4{y}^{2}-25)\)
\(24{y}^{2}-150✓\)

The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.

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

Key Concepts

• How to use a general strategy for factoring polynomials.
  1. Is there a greatest common factor?
     Factor it out.
  2. Is the polynomial a binomial, trinomial, or are there more than three terms?
     If it is a binomial:
     Is it a sum?
     Of squares? Sums of squares do not factor.
     Of cubes? Use the sum of cubes pattern.
     Is it a difference?
     Of squares? Factor as the product of conjugates.
     Of cubes? Use the difference of cubes pattern.
     If it is a trinomial:
     Is it of the form \({x}^{2}+bx+c?\) Undo FOIL.
     Is it of the form \(a{x}^{2}+bx+c?\)
     If a and c are squares, check if it fits the trinomial square pattern.
     Use the trial and error or “ac” method.
     If it has more than three terms:
     Use the grouping method.
  3. Check.
     Is it factored completely?
     Do the factors multiply back to the original polynomial?

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Try it.

\(2{n}^{2}+13n-7\)

Solution

\((2n-1)(n+7)\)

Try it.

\(8{x}^{2}-9x-3\)

Try it.

\({a}^{5}+9{a}^{3}\)

Solution

\({a}^{3}({a}^{2}+9)\)

Try it.

\(75{m}^{3}+12m\)

Try it.

\(121{r}^{2}-{s}^{2}\)

Solution

\((11r-s)(11r+s)\)

Try it.

\(49{b}^{2}-36{a}^{2}\)

Try it.

\(8{m}^{2}-32\)

Solution

\(8(m-2)(m+2)\)

Try it.

\(36{q}^{2}-100\)

Try it.

\(25{w}^{2}-60w+36\)

Solution

\({(5w-6)}^{2}\)

Try it.

\(49{b}^{2}-112b+64\)

Try it.

\({m}^{2}+14mn+49{n}^{2}\)

Solution

\({(m+7n)}^{2}\)

Try it.

\(64{x}^{2}+16xy+{y}^{2}\)

Try it.

\(7{b}^{2}+7b-42\)

Solution

\(7(b+3)(b-2)\)

Try it.

\(30{n}^{2}+30n+72\)

Try it.

\(3{x}^{4}y-81xy\)

Solution

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

Try it.

\(4{x}^{5}y-32{x}^{2}y\)

Try it.

\({k}^{4}-16\)

Solution

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

Try it.

\({m}^{4}-81\)

Try it.

\(5{x}^{5}{y}^{2}-80x{y}^{2}\)

Solution

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

Try it.

\(48{x}^{5}{y}^{2}-243x{y}^{2}\)

Try it.

\(15pq-15p+12q-12\)

Solution

\(3(5p+4)(q-1)\)

Try it.

\(12ab-6a+10b-5\)

Try it.

\(4{x}^{2}+40x+84\)

Solution

\(4(x+3)(x+7)\)

Try it.

\(5{q}^{2}-15q-90\)

Try it.

\(4{u}^{5}+4{u}^{2}{v}^{3}\)

Solution

\(4{u}^{2}(u+v)({u}^{2}-uv+{v}^{2})\)

Try it.

\(5{m}^{4}n+320m{n}^{4}\)

Try it.

\(4{c}^{2}+20cd+81{d}^{2}\)

Solution

prime

Try it.

\(25{x}^{2}+35xy+49{y}^{2}\)

Try it.

\(10{m}^{4}-6250\)

Solution

\(10(m-5)(m+5)({m}^{2}+25)\)

Try it.

\(3{v}^{4}-768\)

Try it.

\(36{x}^{2}y+15xy-6y\)

Solution

\(3y(3x+2)(4x-1)\)

Try it.

\(60{x}^{2}y-75xy+30y\)

Try it.

\(8{x}^{3}-27{y}^{3}\)

Solution

\((2x-3y)(4{x}^{2}+6xy+9{y}^{2})\)

Try it.

\(64{x}^{3}+125{y}^{3}\)

Try it.

\({y}^{6}-1\)

Solution

\((y+1)(y-1)({y}^{2}-y+1)({y}^{2}+y+1)\)

Try it.

\({y}^{6}+1\)

Try it.

\(9{x}^{2}-6xy+{y}^{2}-49\)

Solution

\((3x-y+7)(3x-y-7)\)

Try it.

\(16{x}^{2}-24xy+9{y}^{2}-64\)

Try it.

\({(3x+1)}^{2}-6(3x+1)+9\)

Solution

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

Try it.

\({(4x-5)}^{2}-7(4x-5)+12\)

Try it.

Explain what it mean to factor a polynomial completely.

Solution

Answers will vary.

Try it.

The difference of squares \({y}^{4}-625\) can be factored as \(({y}^{2}-25)({y}^{2}+25).\) But it is not completely factored. What more must be done to completely factor.

Try it.

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

Solution

Answers will vary.

Try it.

Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. outlines a strategy you should use when factoring polynomials.

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example

Try it.

Factor completely: \(4{x}^{5}+12{x}^{4}\).

Solution

\(\begin{array}{lllllll}\text{Is there a GCF?} & & & \text{Yes,}\ 4{x}^{4}. & & & 4{x}^{5}+12{x}^{4} \\ & & & \text{Factor out the GCF.} & & & 4{x}^{4}(x+3) \\ \text{In the parentheses, is it a binomial, a} & & & & & & \\ \text{trinomial, or are there more than three terms?} & & & \text{Binomial.} & & & \\ \text{Is it a sum?} & & & & & & \text{Yes.} \\ \text{Of squares? Of cubes?} & & & & & & \text{No.} \\ \text{Check.} & & & & & & \\ \\ \text{Is the expression factored completely?} & & & & & & \text{Yes.} \\ \text{Multiply.} & & & & & & \\ \\ \\ 4{x}^{4}(x+3) & & & & & & \\ 4{x}^{4}\cdot x+4{x}^{4}\cdot 3 & & & & & & \\ 4{x}^{5}+12{x}^{4}\ ✓ & & & & & & \end{array}\)

Example

Try it.

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

Solution

Is there a GCF?	No.
Is it a binomial, trinomial, or are there more than three terms?	Trinomial.
Are a and c perfect squares?	No, a = 12, not a perfect square.
Use trial and error or the “ac” method. We will use trial and error here.

Check.

\(\ (3x-2)(4x-1)\)

\(\ 12{x}^{2}-3x-8x+2\)

\(\ 12{x}^{2}-11x+2\ ✓\)

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

Key Concepts

• General Strategy for Factoring Polynomials See .
• How to Factor Polynomials
  1. Is there a greatest common factor? Factor it out.
  2. Is the polynomial a binomial, trinomial, or are there more than three terms?
     • If it is a binomial:
       Is it a sum?
       • Of squares? Sums of squares do not factor.
       • Of cubes? Use the sum of cubes pattern.
       Is it a difference?
       
       • Of squares? Factor as the product of conjugates.
       • Of cubes? Use the difference of cubes pattern.
     • If it is a trinomial:
       Is it of the form \({x}^{2}+bx+c\)? Undo FOIL.
       Is it of the form \(a{x}^{2}+bx+c\)?
       • If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
       • Use the trial and error or ‘ac’ method.
     • If it has more than three terms:
       Use the grouping method.
  3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

Try it.

\(10{x}^{4}+35{x}^{3}\)

Solution

\(5{x}^{3}(2x+7)\)

Try it.

\(18{p}^{6}+24{p}^{3}\)

Try it.

\({y}^{2}+10y-39\)

Solution

\((y-3)(y+13)\)

Try it.

\({b}^{2}-17b+60\)

Try it.

\(2{n}^{2}+13n-7\)

Solution

\((2n-1)(n+7)\)

Try it.

\(8{x}^{2}-9x-3\)

Try it.

\({a}^{5}+9{a}^{3}\)

Solution

\({a}^{3}({a}^{2}+9)\)

Try it.

\(75{m}^{3}+12m\)

Try it.

\(121{r}^{2}-{s}^{2}\)

Solution

\((11r-s)(11r+s)\)

Try it.

\(49{b}^{2}-36{a}^{2}\)

Try it.

\(8{m}^{2}-32\)

Solution

\(8(m-2)(m+2)\)

Try it.

\(36{q}^{2}-100\)

Try it.

\(25{w}^{2}-60w+36\)

Solution

\({(5w-6)}^{2}\)

Try it.

\(49{b}^{2}-112b+64\)

Try it.

\({m}^{2}+14mn+49{n}^{2}\)

Solution

\({(m+7n)}^{2}\)

Try it.

\(64{x}^{2}+16xy+{y}^{2}\)

Try it.

\(7{b}^{2}+7b-42\)

Solution

\(7(b+3)(b-2)\)

Try it.

\(3{n}^{2}+30n+72\)

Try it.

\(3{x}^{3}-81\)

Solution

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

Try it.

\(5{t}^{3}-40\)

Try it.

\({k}^{4}-16\)

Solution

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

Try it.

\({m}^{4}-81\)

Try it.

\(15pq-15p+12q-12\)

Solution

\(3(5p+4)(q-1)\)

Try it.

\(12ab-6a+10b-5\)

Try it.

\(4{x}^{2}+40x+84\)

Solution

\(4(x+3)(x+7)\)

Try it.

\(5{q}^{2}-15q-90\)

Try it.

\({u}^{5}+{u}^{2}\)

Solution

\({u}^{2}(u+1)({u}^{2}-u+1)\)

Try it.

\(5{n}^{3}+320\)

Try it.

\(4{c}^{2}+20cd+81{d}^{2}\)

Solution

prime

Try it.

\(25{x}^{2}+35xy+49{y}^{2}\)

Try it.

\(10{m}^{4}-6250\)

Solution

\(10(m-5)(m+5)({m}^{2}+25)\)

Try it.

\(3{v}^{4}-768\)

Try it.

Watermelon drop A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

1. ⓐ The binomial \(-16{t}^{2}+80\) gives the height of the watermelon \(t\) seconds after it is dropped. Factor the greatest common factor from this binomial.
2. ⓑ If the watermelon is thrown down with initial velocity 8 feet per second, its height after \(t\) seconds is given by the trinomial \(-16{t}^{2}-8t+80\). Completely factor this trinomial.

Solution

ⓐ \(-16({t}^{2}-5)\) ⓑ \(-8(2t+5)(t-2)\)

Try it.

Pumpkin drop A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall.

1. ⓐ The binomial \(-16{t}^{2}+128\) gives the height of the pumpkin t seconds after it is dropped. Factor the greatest common factor from this binomial.
2. ⓑ If the pumpkin is thrown down with initial velocity 32 feet per second, its height after \(t\) seconds is given by the trinomial \(-16{t}^{2}-32t+128\). Completely factor this trinomial.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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 completely: \(7{x}^{3}-21{x}^{2}-70x.\)

   Odkryj odpowiedź

   	\(7{x}^{3}-21{x}^{2}-70x\)
   Is there a GCF? Yes, \(7x\).
   Factor out the GCF.	\(7x({x}^{2}-3x-10)\)
   In the parentheses, is it a binomial, trinomial, or are there more terms?
   Trinomial with leading coefficient 1.
   “Undo” FOIL.	\(7x(x\ )(x\ )\)
   	\(7x(x+2)(x-5)\)
   Is the expression factored completely? Yes.
   Neither binomial can be factored.
   Check your answer.
   Multiply.
   \(7x(x+2)(x-5)\)
   \(7x({x}^{2}-5x+2x-10)\)
   \(7x({x}^{2}-3x-10)\)
   \(7{x}^{3}-21{x}^{2}-70x✓\)

2. Factor completely: \(8{y}^{3}+16{y}^{2}-24y.\)

   Odkryj odpowiedź

   \(8y(y-1)(y+3)\)

3. Factor completely: \(5{y}^{3}-15{y}^{2}-270y.\)

   Odkryj odpowiedź

   \(5y(y-9)(y+6)\)

4. Factor completely: \(24{y}^{2}-150.\)

   Odkryj odpowiedź

   	\(24{y}^{2}-150\)
   Is there a GCF? Yes, 6.
   Factor out the GCF.	\(6(4{y}^{2}-25)\)
   In the parentheses, is it a binomial, trinomial or are there more than three terms? Binomial.
   Is it a sum? No.
   Is it a difference? Of squares or cubes? Yes, squares.	\(6({(2y)}^{2}-{(5)}^{2})\)
   Write as a product of conjugates.	\(6(2y-5)(2y+5)\)
   \(\ \text{Is the expression factored completely?}\)
   \(\ \text{Neither binomial can be factored.}\)
   Check:
   \(\ \text{Multiply.}\)
   \(6(2y-5)(2y+5)\)
   \(6(4{y}^{2}-25)\)
   \(24{y}^{2}-150✓\)

5. Factor completely: \(16{x}^{3}-36x.\)

   Odkryj odpowiedź

   \(4x(2x-3)(2x+3)\)

6. Factor completely: \(27{y}^{2}-48.\)

   Odkryj odpowiedź

   \(3(3y-4)(3y+4)\)

7. Factor completely: \(4{a}^{2}-12ab+9{b}^{2}.\)

   Odkryj odpowiedź

   	\(4{a}^{2}-12ab+9{b}^{2}\)
   Is there a GCF? No.
   Is it a binomial, trinomial, or are there more terms?
   Trinomial with \(a\ne 1\). But the first term is a perfect square.
   Is the last term a perfect square? Yes.	\({(2a)}^{2}-12ab+{(3b)}^{2}\)
   Does it fit the pattern, \({a}^{2}-2ab+{b}^{2}\)? Yes.	\({(2a)}^{2}{}_{\text{↘}}\underset{-2(2a)(3b)}{-12ab+}{}_{\text{↙}}{(3b)}^{2}\)
   Write it as a square.	\({(2a-3b)}^{2}\)
   \(\ \text{Is the expression factored completely? Yes.}\)
   \(\ \text{The binomial cannot be factored.}\)
   Check your answer.
   \(\ \text{Multiply.}\)
   \({(2a-3b)}^{2}\)
   \({(2a)}^{2}-2\cdot 2a\cdot 3b+{(3b)}^{2}\)
   \(4{a}^{2}-12ab+9{b}^{2}✓\)

8. Factor completely: \(4{x}^{2}+20xy+25{y}^{2}.\)

   Odkryj odpowiedź

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

9. Factor completely: \(9{x}^{2}-24xy+16{y}^{2}.\)

   Odkryj odpowiedź

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

10. Factor completely \(12{x}^{3}{y}^{2}+75x{y}^{2}.\)

    Odkryj odpowiedź

    	\(12{x}^{3}{y}^{2}+75x{y}^{2}\)
    Is there a GCF? Yes, \(3x{y}^{2}\).
    Factor out the GCF.	\(3x{y}^{2}(4{x}^{2}+25)\)
    In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial.
    Is it a sum? Of squares? Yes.	Sums of squares are prime.
    \(\ \text{Is the expression factored completely? Yes.}\)
    Check:
    \(\ \text{Multiply.}\)
    \(3x{y}^{2}(4{x}^{2}+25)\)
    \(12{x}^{3}{y}^{2}+75x{y}^{2}✓\)

11. Factor completely: \(50{x}^{3}y+72xy.\)

    Odkryj odpowiedź

    \(2xy(25{x}^{2}+36)\)

12. Factor completely: \(27x{y}^{3}+48xy.\)

    Odkryj odpowiedź

    \(3xy(9{y}^{2}+16)\)

13. Factor completely: \(24{x}^{3}+81{y}^{3}.\)

    Odkryj odpowiedź

    Is there a GCF? Yes, 3.
    Factor it out.
    In the parentheses, is it a binomial, trinomial, of are there more than three terms? Binomial.
    Is it a sum or difference? Sum.
    Of squares or cubes? Sum of cubes.
    Write it using the sum of cubes pattern.
    Is the expression factored completely? Yes.
    Check by multiplying.

14. Factor completely: \(250{m}^{3}+432{n}^{3}.\)

    Odkryj odpowiedź

    \(2(5m+6n)(25{m}^{2}-30mn+36{n}^{2})\)

15. Factor completely: \(2{p}^{3}+54{q}^{3}.\)

    Odkryj odpowiedź

    \(2(p+3q)({p}^{2}-3pq+9{q}^{2})\)

16. Factor completely: \(3{x}^{5}y-48xy.\)

    Odkryj odpowiedź

    	\(3{x}^{5}y-48xy\)
    Is there a GCF? Factor out \(3xy\)	\(3xy({x}^{4}-16)\)
    Is the binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	\(3xy({({x}^{2})}^{2}-{(4)}^{2})\)
    Factor it as a product of conjugates	\(3xy({x}^{2}-4)({x}^{2}+4)\)
    The first binomial is again a difference of squares.	\(3xy({(x)}^{2}-{(2)}^{2})({x}^{2}+4)\)
    Factor it as a product of conjugates.	\(3xy(x-2)(x+2)({x}^{2}+4)\)
    Is the expression factored completely? Yes.
    Check your answer.
    Multiply.
    \(3xy(x-2)(x+2)({x}^{2}+4)\)
    \(3xy({x}^{2}-4)({x}^{2}+4)\)
    \(3xy({x}^{4}-16)\)
    \(3{x}^{5}y-48xy✓\)

17. Factor completely: \(4{a}^{5}b-64ab.\)

    Odkryj odpowiedź

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

18. Factor completely: \(7x{y}^{5}-7xy.\)

    Odkryj odpowiedź

    \(7xy({y}^{2}+1)(y-1)(y+1)\)

19. Factor completely: \(4{x}^{2}+8bx-4ax-8ab.\)

    Odkryj odpowiedź

    	\(4{x}^{2}+8bx-4ax-8ab\)
    Is there a GCF? Factor out the GCF, 4.	\(4({x}^{2}+2bx-ax-2ab)\)
    There are four terms. Use grouping.	\(\begin{array}{l} \\ 4[x(x+2b)-a(x+2b)] \\ 4(x+2b)(x-a)\end{array}\)
    Is the expression factored completely? Yes.
    Check your answer. Multiply. \(\ \begin{array}{l} \\ \\ 4(x+2b)(x-a) \\ 4({x}^{2}-ax+2bx-2ab) \\ 4{x}^{2}+8bx-4ax-8ab✓\end{array}\)

20. Factor completely: \(6{x}^{2}-12xc+6bx-12bc.\)

    Odkryj odpowiedź

    \(6(x+b)(x-2c)\)

21. Factor completely: \(16{x}^{2}+24xy-4x-6y.\)

    Odkryj odpowiedź

    \(2(4x-1)(2x+3y)\)

22. Factor completely: \(40{x}^{2}y+44xy-24y.\)

    Odkryj odpowiedź

    	\(40{x}^{2}y+44xy-24y\)
    Is there a GCF? Factor out the GCF, \(4y\).	\(4y(10{x}^{2}+11x-6)\)
    Factor the trinomial with \(a\ne 1\).	\(4y(10{x}^{2}+11x-6)\)
    	\(4y(5x-2)(2x+3)\)
    Is the expression factored completely? Yes.
    Check your answer.
    Multiply.
    \(4y(5x-2)(2x+3)\)
    \(4y(10{x}^{2}+11x-6)\)
    \(40{x}^{2}y+44xy-24y✓\)

23. Factor completely: \(4{p}^{2}q-16pq+12q.\)

    Odkryj odpowiedź

    \(4q(p-3)(p-1)\)

24. Factor completely: \(6p{q}^{2}-9pq-6p.\)

    Odkryj odpowiedź

    \(3p(2q+1)(q-2)\)

25. Factor completely: \(9{x}^{2}-12xy+4{y}^{2}-49.\)

    Odkryj odpowiedź

    	\(9{x}^{2}-12xy+4{y}^{2}-49\)
    Is there a GCF? No.
    With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms.	\(9{x}^{2}-12xy+4{y}^{2}-49\)
    Factor the trinomial with \(a\ne 1\). But the first term is a perfect square.
    Is the last term of the trinomial a perfect square? Yes.	\({(3x)}^{2}-12xy+{(2y)}^{2}-49\)
    Does the trinomial fit the pattern, \({a}^{2}-2ab+{b}^{2}\)? Yes.	\({(3x)}^{2}{}_{\text{↘}}\underset{-2(3x)(2y)}{-12xy+}{}_{\text{↙}}{(2y)}^{2}-49\)
    Write the trinomial as a square.	\({(3x-2y)}^{2}-49\)
    Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	\({(3x-2y)}^{2}-{7}^{2}\)
    Write it as a product of conjugates.	\(((3x-2y)-7)((3x-2y)+7)\)
    	\((3x-2y-7)(3x-2y+7)\)
    Is the expression factored completely? Yes.
    Check your answer.
    Multiply.
    \((3x-2y-7)(3x-2y+7)\)
    \(9{x}^{2}-6xy-21x-6xy+4{y}^{2}+14y+21x-14y-49\)
    \(9{x}^{2}-12xy+4{y}^{2}-49✓\)

26. Factor completely: \(4{x}^{2}-12xy+9{y}^{2}-25.\)

    Odkryj odpowiedź

    \((2x-3y-5)(2x-3y+5)\)

27. Factor completely: \(16{x}^{2}-24xy+9{y}^{2}-64.\)

    Odkryj odpowiedź

    \((4x-3y-8)(4x-3y+8)\)

28. \(2{n}^{2}+13n-7\)

    Odkryj odpowiedź

    \((2n-1)(n+7)\)

29. \(8{x}^{2}-9x-3\)

30. \({a}^{5}+9{a}^{3}\)

    Odkryj odpowiedź

    \({a}^{3}({a}^{2}+9)\)

31. \(75{m}^{3}+12m\)

32. \(121{r}^{2}-{s}^{2}\)

    Odkryj odpowiedź

    \((11r-s)(11r+s)\)

33. \(49{b}^{2}-36{a}^{2}\)

34. \(8{m}^{2}-32\)

    Odkryj odpowiedź

    \(8(m-2)(m+2)\)

35. \(36{q}^{2}-100\)

36. \(25{w}^{2}-60w+36\)

    Odkryj odpowiedź

    \({(5w-6)}^{2}\)

37. \(49{b}^{2}-112b+64\)

38. \({m}^{2}+14mn+49{n}^{2}\)

    Odkryj odpowiedź

    \({(m+7n)}^{2}\)

39. \(64{x}^{2}+16xy+{y}^{2}\)

40. \(7{b}^{2}+7b-42\)

    Odkryj odpowiedź

    \(7(b+3)(b-2)\)

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: General Strategy for Factoring Polynomials

1. Recognize and use the appropriate method to factor a polynomial completely
2. Is there a greatest common factor?
3. Is the polynomial a binomial, trinomial, or are there more than three terms?
4. Is it a sum?
5. Is it a difference?
6. Is it of the form
7. Is it of the form
8. Use the grouping method.

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.

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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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