Factor Second Degree Polynomials Vaughan ON

Steps

1. Set up your equation. Order the numbers from highest to lowest power and then factor out the greatest common factor if one exists.6 + 6x2 + 13x6x2 + 13x + 6
2. Find the factored form using one of the methods below.(2x + 3)(3x + 2)
3. Check your work by multiplying out the factors using FOIL. Then combine like terms and you're done!(2x + 3)(3x + 2)6x2 + 4x + 9x + 66x2 + 13x + 6

Trial and Error Method If you have a fairly simple equation you'll be able to figure out the factors yourself. Example: 3x2 + 2x - 8

1. List the factors of the a term and the c term.a = 3 factors: 1 and 3c = -8 factors: 2 and 4 or 1 and 8
2. Write down two sets of parentheses with empty spaces like this:( x    )( x    )
3. Fill 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   )
4. Fill 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).
5. Decide 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)
6. Test 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 - 83x2 + 5x - 8 3x2 + 2x - 8
7. Swap 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.
8. Reverse 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.
9. Double-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

1. Multiply the a term (6 in the example) by the c term (also 6 in the example).6 6 = 36
2. Find two numbers that when multiplied equal this number (36) and add up to be the b term (13).4 9 = 36   4 + 9 = 13
3. Substitute the two numbers you get into this form as k and h (order doesn't matter): ax2 + kx + hx + c6x2 + 4x + 9x + 6
4. Factor 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 + 62x(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

1. Multiply the a term (8 in the example) by the c term (2 in this example).8 2 = 16
2. Find 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
3. Take 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
4. Look 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)----------------------     8Answer:(x + 1)(8x + 2)
5. Take the GCF (if any) out of either or both parentheses.(x + 1)(8x + 2)2(x + 1)(4x + 1)

Difference of Two Squares

1. Factor out a GCF if you need to.27x2 - 123(9x2 - 4)
2. Decide 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. (9x2) = 3x and (4) = 2 (notice that we have left out the negative sign)
3. Put the a and c values from your equation into this expression:( (a) + (c))( (a) - (c))3[( (9x2) + (4))( (9x2) - (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

1. Plug the corresponding values into the quadratic formula:x = -b (b2 - 4ac)      ---------------------                2ax = -4 (42 - 4 1 1)      -----------------------                  2 1
2. Solve for x. You should get two x values.x= -4 (16 - 4)     ------------------              2x = -4 (12)      --------------              2x = -4 (4 3)      --------------              2x = -4 2 (3)      --------------              2x = -2 (3)x = -2 + (3) or x = -2 - (3)
3. Plug the x values (h and k) into this expression: (x - h)(x - k)(x - (-2 + (3))(x - (-2 - (3))(x + 2 + (3))(x + 2 - (3))

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

1. Enter your equation into the [Y = ] screen.y = x2 x 2
2. Press [GRAPH]. You should see a smooth arc.
3. Locate 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.
4. Plug the x values (h and k) into this expression: (x - h)(x - k)(x - (-1))(x - 2)(x + 1)(x - 2)

Tips

• Eventually you will be able to do trial and error in your head. Until then make sure to write it out however!
• If a term has no coefficient as written, the coefficient is one.x2 = 1x2
• If a term does not exist the coefficient is 0. It will be helpful to rewrite the equation if this occurs.x2 + 6 = x2 + 0x + 6
• If you factored your equation 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 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.

Warnings

• If 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.

Things You'll Need

• Pencil
• Paper
• Quadratic equation (also called a 2nd degree polynomial)
• Graphing calculator (optional)

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