How to Factor Second Degree Polynomials (Quadratic Equations)

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Pin ItArticle EditDiscussThis will instruct you on how to factor second degree polynomials . A polynomial contains a variable (x) raised to a power, known as a degree, and several terms and/or constants. To factor a polynomial means to break the expression down into more manageable chunks that are multiplied together. These skills are Algebra I and above, and therefore may be difficult to understand if your math skills are not at this level.

Throughout this article the terms will be referred to according to the standard form of a quadratic equation:

ax2 + bx + c = 0

Edit Steps1Set up your expression. Order the numbers from highest to lowest power and then factor out the greatest common factor if one exists.

6 + 6x2 + 13x

6x2 + 13x + 62Find the factored form using one of the methods below.

(2x + 3)(3x + 2)

3Check your work by multiplying out the factors using FOIL. Then combine like terms and you're done!

(2x + 3)(3x + 2)

6x2 + 4x + 9x + 6

6x2 + 13x + 6

Trial and Error Method

If you have a fairly simple polynomial, you'll be able to figure out the factors yourself. NOTE: Using this method may not be as simple as when factoring more complicated trinomials.

Example: 3x2 + 2x - 8

1List the factors of the a term and the c term.

a = 3 factors: 1 and 3

c = -8 factors: 2 and 4 or 1 and 8

2Write down two sets of parentheses with empty spaces like this:

( x    )( x    )

3Fill the spaces in front of the x's with a pair of possible factors of the a value. There is only one possibility for our example:

(3x   )(1x   )

4Fill in the two spaces after the x's with a pair of factors for the constant. Let's say we choose (3x  8)(x  1).

5Decide what signs should be between the x's and the numbers. Here's a guide:

If ax2 + bx + c then (x + h)(x + k)

If ax2 - bx - c or ax2 + bx - c then (x - h)(x + k)

If ax2 - bx + c then (x - h)(x - k)

For our example 3x2 + 2x - 8 so (x - h)(x + k)

We'll have to guess as for the rest. (3x + 8)(x - 1)

6Test your choice by multiplying (use FOIL) the two parentheses together. If the middle term is not at least the correct value (disregarding positive or negative) you have chosen the wrong c factors.

(3x + 8)(x - 1)

3x2 - 3x + 8x - 8

3x2 + 5x - 8 ? 3x2 + 2x - 8

7Swap out your choices if necessary. In our example, let's try 2 and 4 instead of 1 and 8: (3x + 2)(x - 4)

Now our c term is an -8.But our Outside/Inside combo is -12x and 2x, which will not combine to make the correct b term of +2x.8Reverse the order if necessary. Let's try moving the 2 and 4 around: (3x + 4)(x - 2)

c term is still okay.Outside/Inside combo is -6x and 4x. If we combine them, we get pretty close to the 2x we were aiming for --- right amount, wrong sign.9Double-check your signs if necessary. We're going to stick with the same order, but swap which one has the subtraction: (3x - 4)(x + 2)

c term is still okay.Outside/Inside combo is now 6x and -4x. This will combine to create the positive 2x from the original problem, so these are the correct factors.

Decomposition Method

If the numbers are large or you're just tired of guesswork use this method.

Example: 6x2 + 13x + 6

1Multiply the a term (6 in the example) by the c term (also 6 in the example).

6•6 = 36

2Find two numbers that when multiplied equal this number (36) and add up to be the b term (13).

4•9 = 36   4 + 9 = 13

3Substitute the two numbers you get into this form as k and h (order doesn't matter): ax2 + kx + hx + c

6x2 + 4x + 9x + 6

4Factor the polynomial by grouping. Organize the equation so that you can take out the greatest common factor of the first two terms and the last two terms. Both factored groups should be the same. Add the GCF's together and enclose them in parentheses next to the factored group.

6x2 + 4x + 9x + 6

2x(3x + 2) + 3(3x + 2)

(2x + 3)(3x + 2)

Triple Play Method

It is very similar to the decomposition method but it is simpler.

Example: 8x2 + 10x + 2

1Multiply the a term (8 in the example) by the c term (2 in this example).

8•2 = 16

2Find the two numbers whose product is this number (16) and whose sum is equal to the b term (10).

2•8 = 16   8 + 2 = 10

3Take these two numbers (which we will call h and k) and substitute them into this expression:

(ax + h)(ax + k)
----------------------
     a

(8x + 8)(8x + 2)
----------------------
     8

4Look to see which one of the two parenthesis terms in the numerator is evenly divisible by a {in this example it is (8x + 8)}. Divide this term by a and leave the other one as is.

(8x + 8)(8x + 2)
----------------------
     8

Answer:(x + 1)(8x + 2)

5Take the GCF (if any) out of either or both parentheses.

(x + 1)(8x + 2)

2(x + 1)(4x + 1)

Difference of Two Squares1Factor out a GCF if you need to.

27x2 - 12

3(9x2 - 4)

2Decide if your equation is a difference of squares. It must have two terms and you should be able to take the square root of the terms evenly.

v(9x2) = 3x and v(4) = 2 (notice that we have left out the negative sign)

3Put the a and c values from your equation into this expression:

(v(a) + v(c))(v(a) - v(c))

3[(v(9x2) + v(4))(v(9x2) - v(4))]

3[(3x + 2)(3x - 2)]

Using the Quadratic Formula

If all else fails and the equation will not factor evenly use the quadratic formula.





Example: x2 + 4x + 1

1Plug the corresponding values into the quadratic formula:

x = -b ± v(b2 - 4ac)
      ---------------------
                2a

x = -4 ± v(42 - 4•1•1)
      -----------------------
                  2•1

2Solve for x. You should get two x values.

x= -4 ± v(16 - 4)
     ------------------
              2

x = -4 ± v(12)
      --------------
              2

x = -4 ± v(4•3)
      --------------
              2

x = -4 ± 2v(3)
      --------------
              2

x = -2 ± v(3)

x = -2 + v(3) or x = -2 - v(3)

3Plug the x values (h and k) into this expression: (x - h)(x - k)

(x - (-2 + v(3))(x - (-2 - v(3))

(x + 2 + v(3))(x + 2 - v(3))

Using a Calculator

These directions are for a TI graphing calculator. These are especially useful in standardized tests.

1Enter your equation into the [Y = ] screen.

y = x2 - x - 2

2Press [GRAPH]. You should see a smooth arc.

3Locate where the arc intersects the x axis. These are the x values.

(-1, 0), (2 , 0)

x = -1, x = 2

If you cannot identify them by sight press [2nd] and then [TRACE]. Press [2] or select "zero". Slide the cursor to the left of an intersect and press [ENTER]. Slide the cursor to the right of an intersect and press [ENTER]. Slide the cursor as close as possible to the intersect and press [ENTER]. The calculator will find the x value. Do this for the other intersect also.4Plug the x values (h and k) into this expression: (x - h)(x - k)

(x - (-1))(x - 2)

(x + 1)(x + (-2)) great plus add it to the common factor

Edit Video

Edit TipsIf you have a TI-84 calculator (graphing) there is a program named SOLVER that will solve a quadratic equation. It will also solve any other degree polynomial.If you factored your polynomial using the quadratic formula and got an answer with a radical, you may want to convert the x values to fractions in order to check it.If a term has no coefficient as written, the coefficient is 1.

x2 = 1x2If a term does not exist the coefficient is 0. It will be helpful to rewrite the equation if this occurs.

x2 + 6 = x2 + 0x + 6Eventually you will be able to do trial and error in your head. Until then, make sure to write it out.

Edit WarningsIf you are learning this concept in a math class, pay attention to what your teacher advises and do not just use your favorite method. Your teacher may ask you to use a specific method on the test or not allow graphing calculators.

Edit Things You'll NeedPencilPaperQuadratic equation (also called a 2nd degree polynomial)Graphing calculator (optional)

Edit Related wikiHowsHow to Graph a Quadratic EquationHow to Factor TrinomialsHow to Factor a NumberHow to Calculate PercentagesHow to Differentiate PolynomialsHow to Do Math ProofsHow to Calculate Your Age by ChocolateHow to Add 5 Consecutive Numbers QuicklyHow to Solve Quadratic EquationsHow to Find the Signs of a,b,c by Seeing the Graph of a Quadratic Polynomial

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