Content

• Steps
• Part 1 of 3: Defining a Method
• Part 2 of 3: Factoring the Divisible
• Part 3 of 3: Long Division
• Tips
• Warnings

Polynomials can be divided in the same way as numbers: either by factoring or by long division. The method used depends on the type of the polynomial and the type of the divisor.

Steps

Part 1 of 3: Defining a Method

1. 1 Determine the type of divider. The divisor (the polynomial you are dividing by) is compared with the dividend (the polynomial you are dividing) and the appropriate division method is determined.
   • If the divisor is a monomial, which is a coefficient of a variable or an intercept (coefficient without a variable), you can probably factor the divisor and cancel one of the factors and the divisor. See the section "Factoring a Divisible".
   • If the divisor is binomial (a polynomial with two terms), you can probably factor the dividend and cancel one of the factors and the divisor.
   • If the divisor is a trinomial (a polynomial with three terms), you can probably factor both the dividend and the divisor and then cancel the common factor or long division.
   • If the divisor is a polynomial with more than three terms, you will most likely have to use long division. See the Long Division section.
2. 2 Determine the type of dividend. If the type of the divisor does not tell you the method of division, determine the type of the dividend.
   • If the dividend has three or fewer terms, you can probably factor the dividend and cancel one of the factors and the divisor.
   • If the dividend has more than three members, you will most likely need to use long division.

Part 2 of 3: Factoring the Divisible

1. 1 Find the common factor for the divisor and the dividend. If it exists, you can bracket it out and shorten it.
   • Example. When dividing 3x - 9 by 3 in a binomial, put 3 outside the brackets: 3 (x - 3). Then cancel the outside parentheses 3 and the divisor (3). Answer: x - 3.
   • Example: When dividing 24x - 18x by 6x in a binomial, put 6x outside the brackets: 6x (4x - 3). Then cancel the parentheses 6x and the divisor (6x). Answer: 4x - 3.
2. 2 Determine if the dividend can be factorized using abbreviated multiplication formulas. If one of the factors is equal to the divisor, then you can cancel them. Here are some formulas for abbreviated multiplication:
   • Difference of squares. It is a binomial of the form ax - b, where the values ​​of a and b are perfect squares (that is, you can extract the square root of these numbers). This binomial can be decomposed into two factors: (ax + b) (ax - b).
   • Full square. This is a trinomial of the form ax + 2abx + b, which can be decomposed into two factors: (ax + b) (ax + b) or written as (ax + b). If the second term is preceded by a minus, this trinomial is expanded as: (ax - b) (ax - b).
   • Sum or difference of cubes. It is a binomial of the form ax + b or ax - b, where the values ​​of a and b are complete cubes (that is, you can extract the cube root from these numbers). The sum of cubes is decomposed into: (ax + b) (ax - abx + b). The difference between the cubes is decomposed into: (ax - b) (ax + abx + b).
3. 3 Use trial and error to factor the dividend. If you see that the abbreviated multiplication formula cannot be applied to the dividend, try expanding the dividend in other ways. First, find the factors of the intercept, taking into account the coefficient of the second term of the dividend.
   • Example. If the dividend is x - 3x - 10, find the factors of the intercept 10, taking into account the factor 3.
   • The number 10 can be split into the following factors: 1 and 10 or 2 and 5. Since there is a minus in front of 10, a minus must also appear in front of one of the factors of 10.
   • The coefficient 3 is 5-2, so we choose the factors 5 and 2. Since there is a minus in front of 3, there must also be a minus in front of 5. Thus, the dividend is decomposed into factors: (x - 5) (x + 2). If the divisor is equal to one of these two factors, then they can be canceled.

Part 3 of 3: Long Division

1. 1 Write down the dividend and divisor in the same way as you write down ordinary numbers when they are divided into a column.
   • Example. Divide x + 11 x + 10 by x +1.
2. 2 Divide the first term of the dividend by the first term of the divisor. Write down the result.
   • Example. Divide x (the first term of the dividend) by x (the first term of the divisor). Write down the result: x.
3. 3 Multiply the result from the previous step (x) by the divisor. Write the multiplication result under the first and second terms of the dividend, respectively.
   • Example. Multiply x by x + 1 to get x + x. Write this binomial under the first and second terms of the dividend, respectively.
4. 4 Subtract the result (from the previous step) from the dividend. First of all, subtract the multiplication result (obtained in the previous step) from the dividend, and then remove the free term.
   • Reverse the signs of the binomial x + x and write it as - x - x. Subtracting this binomial from the first two terms in the dividend gives 10x. After demolishing the free term of the dividend, you will get a binomial 10x + 10 (intermediate binomial).
5. 5 Repeat the previous three steps with the intermediate binomial (obtained in the previous step). You will divide its first term by the first term of the divisor and write the result next to the result of the first division. Then multiply this second division result by the divisor and subtract the result of the multiplication from the intermediate binomial.
   • Since 10x / x = 10, write "+10" after the result of the first division (x).
   • Multiplying 10 by x +1, you get the binomial 10x + 10. Change the signs of this binomial (- 10x - 10) and write it down under the intermediate binomial accordingly.
   • Subtract the binomial obtained in the previous step from the intermediate binomial and you get 0. So x + 11 x + 10 divided by x +1 is x + 10 (you may get the same result by factoring the trinomial, but this trinomial was chosen as the simplest example).

Tips

• If you get a remainder after long division, you can write it down as a fractional term with the remainder in the numerator and the divisor in the denominator. For example, if instead of x + 11 x + 10 you are given x + 11 x + 12, then dividing this trinomial by x + 1 you get the remainder 2. Therefore, write the answer (quotient) in the form: x + 10 + (2 / ( x +1)).
• If a given polynomial does not have a member with a variable of the appropriate order, for example, 3x + 9x + 18 does not have a member with a variable of the first order, you can add the missing term with a coefficient of 0 (in our example, it is 0x) to correctly position the terms during division. This move will not change the value of this polynomial.

Warnings

• When division in a column, write the terms correctly (write terms of the same order under each other) to avoid errors when subtracting terms.
• When writing a division result that includes a fractional term, always precede the fractional term with a plus sign.