Products and factors of polynomial, Synthetic division
Factoring polynomials:
Factoring polynomials with a degree greater than 2 is almost the same as factoring a quadratic expression. In a quadratic expression you will write it out as a product of 2 factors, but with a polynomial expression of a degree greater than 2 you will write it out as a product of more than 2. For example x^3-5x^2-6x is a polynomial expression with a degree of 3. So you first find out what all the numbers have in common which is x. So you factor out an x from all the numbers. You end up getting x(x^2-5x-6). Then you factor out what is remaining. You get x(x-6) (x+1).
Practice: factor each polynomial
- x^3+8x^2+15x
- x^3+6x^2+8x
- x^3-x^2-30x
Polynomials can also be factored in pairs
Sometimes you won’t be able to factor out a number or variable that satisfies all of the terms and that is when you have to group them together and solve them separately. For example x^3+4x^2+2x+8, you can’t get something that satisfies all of the terms. So group them in pairs. (x^3+4x^2)+(2x+8). After you have done that you can factor something out of each. x^2 can be factored out of (x^3+4x^2). So it will turn into x^2(x+4). Do the same from (2x+8) and you will get 2(x+4). Then since both terms have a (x+4) you factor it out of each term. So your final answer will be (x^2+2)(x+4).
Practice: Factor each polynomial
- x^3+3x^2-2x-6
- x^3-2x^2-5x+10
Factoring the sum and difference of two cubes
You must memorize these two equations if you are trying to find the sum or different of two cubes. If I gave you the equation x^3+27 follow the first equation to solve. First distinguish which one is a and b. A is x and you have to do a little work to find b. You have to find out what to the third power will equal 27, that is 3. So now you have a and b and can proceed onto the next part. All you have to do now is sub in what a and b are into the equation. (x+3)(x^2-3x+3^2). You can simplify that into (x+3)(x^2-3x+9).
Practice:
- x^3-64
- x^3-216
Synthetic division
Synthetic division can be used to divide a polynomial only by a linear binomial of the form x-r.
So if you had x-2 being the divisor and x^3+3x^2-4x-12 being the dividend you would first write the coefficients of the polynomial x^3+3x^2-4x-12 in a horizontal line. Then you would write the r value which is 2 before your first coefficient and put a square around the two. So it would look like this:
2_| 1 3 -4 -12
______
Then write the first coefficient which is 1 below the line
2_| 1 3 -4 -12
______
1
Then you multiply the r value by the number below the line and write the product below the next coefficient, but not below the line.
2_| 1 3 -4 -12
____2______
1
Then add 3 and 2 together and write the answer below the line.
2_| 1 3 -4 -12
____2______
15
Do the same thing and multiply the and the 5 then write the product below the next coefficient, add them together and continue. You will end up with:
2_| 1 3 -4 -12
____2_10_12__
1 5 6 0
After you are finished the resulting numbers are 1,5 and 6 are the coefficients of the quotient, x^2+5x+6.
Answers to practice
- x(x+3)(x+5)
- x(x+4)(x+2)
- x(x-6)(x+5)
- (x^2-2)(x+3)
- (x^2-5)(x-2)
- (x+3)(x^2-3x+9)
- (x-6)(x^2+6x+36)