Algebra 2- How do you factor 16x^3+54?
I've got my algebra two final tomorrow and I kind of suck at these things. The review sheet says the answer is 2(2x+3)(4x^2-6x-9), but I don't understand how you get there.
Update:thank you so much!!!! i really appreciate it
Comments
Hi,
I'll give an example and then do your problem.
Here are rules for the sum and difference of perfect cubes. First, always look first for a GCF. If there is one, factor it out. Then, perfect cube problems will always factor into a binomial times a trinomial, like this:
x³ + 125 = (x + 5)(x² - 5x + 25)
The general rule for these is:
1) Factor out a GCF if there is one.
2) Make 2 parentheses for a binomial and a trinomial.
3) The sign from the problem goes in the binomial. The opposite sign from what you just used goes in the trinomial, and is always followed by a "+" as the second sign in the trinomial. This is the only step where what you do is different on the sum of cubes from on the difference of cubes.
4) Figure out what you would cube to get the first term of the problem and put that expression as the first term of your binomial. Then figure out what you would cube to get the second term of the problem and put that expression as the second term of your binomial.
5) Your trinomial already has its signs, so don't worry about the signs. To find terms of the trinomial, write the first term of your binomial 3 times followed by the second term of your binomial 3 times out to the side. Multiply the first 2 things together to get the trinomial's first term. Multiply the next 2 things together to get the trinomial's middle term. Multiply the last 2 things together to get the trinomial's third term.
So for, x³ + 125, there is no GCF.
Parentheses would be ( + )( - + )
To get x³, you'd cube x.
To get 125, you'd cube 5.
So your binomial becomes (x + 5)( - + )
Now write x x x 5 5 5
x*x = x²
x*5 = 5x
5*5 = 25
So your trinomial becomes (x + 5)(x² - 5x + 25)
Here's your problem:
1) In 16x³ + 54, look for a GCF. This time 2 factors out of each term, giving:
2(8x³ + 27)
2) Make 2 parentheses for a binomial and a trinomial.
3) The sign from the problem goes in the binomial. The opposite sign from what you just used goes in the trinomial, and is always followed by a "+" as the second sign in the trinomial.
Parentheses would be ( + )( - + )
4) Figure out what you would cube to get the first term of the problem and put that expression as the first term of your binomial. Then figure out what you would cube to get the second term of the problem and put that expression as the second term of your binomial.
To get 8x³, you'd cube 2x.
To get 27, you'd cube 3.
So your binomial becomes (2x + 3)( - + )
5) Your trinomial already has its signs, so don't worry about the signs. To find terms of the trinomial, write the first term of your binomial 3 times followed by the second term of your binomial 3 times out to the side. Multiply the first 2 things together to get the trinomial's first term. Multiply the next 2 things together to get the trinomial's middle term. Multiply the last 2 things together to get the trinomial's third term.
Now write 2x 2x 2x 3 3 3
2x*2x = 4x²
2x*3 = 6x
3*3 = 9
So your trinomial becomes 2(2x + 3)(4x² - 6x + 9)
16x³ + 54 = 2(2x + 3)(4x² - 6x + 9) <==YOUR ANSWER
I hope that helps you! Do well on your final!! :-)
First, you factor out a 2 from both terms:
2(8x^3 + 27). Now in the parentheses, you have the sum of two cubes. Remember the addition of two cubes:
2(2x + 3)(4x^2 - 6x - 9).
-John
First factor out a 2 to get 8x^3+27. This is now a sum of cubes therefore, 8x^3 is 2x and the cube of 27 is three. (2x+3), now factor the rest to be2 (2x+3)(4x^2-6x-9) the standard form of the sum of cubes is (a + b)(a^2 - 2b - b^2)
you'll be able to desire to apply long branch to divide x-6 into x^3-10x^2+19x+30. in case you do this top, you will get x^2+4x-5. This factors into x+5 and x-a million, you get (x-6) x (x+5) x (x-a million) in case you do not understand a thank you to do the polynomial long branch or a thank you to element the binomial, ask a stick to up question. i will quickly provide the uncomplicated concept of each: First, to do the long branch, you notice that X-6 leads to six and the given polynomial leads to 30. 6 circumstances -5 is 30. so which you start up off with '-5'. x-6 circumstances -5 is -5x+30. Subtracting you get X^3-10x^2+24x. -6 is going into 24x 4x circumstances. 4x circumstances x-6 is 4x^2-24x, subtracting provides x^3-6x^2. -6 is going into 6^2 circumstances, providing you with an exceptionally final answer of x^2+4x-5. To element x^2+4x-5, merely see that 5 is 5 circumstances a million and that 4x is constructive. To get '-5' you will might desire to distinctive a detrimental extensive form by capability of a favorable, and because 4 is constructive, the better extensive form might desire to be constructive (so their sum would be postive). for that reason x+5 circumstances x-a million.