Before learning how to use the factoring trinomials calculator, let’s discuss what the trinomials are and various methods of their factoring.

Factoring polynomials is the representation of the polynomial as a product of polynomials of lower degrees.

According to the fundamental algebraic theorem each polynomial over the complex numbers can be represented as a product of linear polynomials, including the way up to a constant factor, and the order of the following of the factors.

The opposite of factoring polynomials is their extension - multiplication of polynomial factors to obtain the expanded polynomial written as a sum of terms.

In algebra, the trinomial is a polynomial represented as the sum of the three components, or a special condition. The most famous type of trinomial is quadratic trinomial – ax^{2} + bx + c = a. However, not all trinomials are quadratic. Some may have a slightly higher degree of variables or degrees. The polynomials have a number of applications in mathematics and science, and knowing how to factor trinomials (manually or with the use of factoring trinomials calculator) can be applied in many areas that require algebraic skills. Below, there are the steps one needs to follow when factoring trinomials. There are several special cases, when trinomials can be expanded. If none of them is used, it may be necessary to use general methods of factoring of polynomials with higher degree.

The first method of factoring trinomials is to use the factoring trinomials calculator, which can be downloaded or used online on various websites. When you open a page with the factoring trinomials calculator, you need to enter an expression you need to factor. The calculator will simplify the expression by expanding multiplication and putting together the same terms. At this point, the calculator will try to factor the expression by dividing a GCF, and determining a difference between two squares, or factorable trinomials. The following rules will assist you to put the expression into the factoring trinomials calculator:

- Variables.

Any letter of a lowercase may be used as a variable. - Exponents.

Exponents are backed up on variables using the caret symbol in factoring trinomials calculator. For example, to express x^{2}, you need to press x and caret symbol and then 2. Keep in mind that exponents must be positive integers, no negatives, variables, or decimals. Exponents may not currently be put on numbers, parentheses, or brackets. - Brackets and parentheses.

Brackets [] and parentheses () can be used to group terms as in a usual expression. - Addition, multiplication, and subtraction.

On factoring trinomials calculator, for subtraction and addition use the regular «-» and «+» symbols respectively. For multiplication use the «*» symbol, which is optional when multiplying a number by a variable. For example, 2 * x can also be input as 2x. The same applies to the following: 2 * (x + 5), which can also be inserted as 2(x + 5); 2x * (5) can be input as 2x(5). The «*» is also optional when multiplying parentheses, for instance, (x + 1)(x - 1). - Order of operations.

The factoring trinomials calculator uses the regular order of operations taught by most algebra books – parentheses, exponents, multiplication and division, addition, and subtraction. The only exclusion is that division is not currently supported in the calculator. Attempts to use the «/» symbol will end in an error. - Square root, division, radicals, fractions.

Square root, division, radicals, and fractions are not supported on factoring trinomials calculator at the moment.

The other methods of factoring represented below are the ones that can be applied without the factoring trinomials calculator.

Factoring x^{2} + bx + c

Learn how to multiply two binomials members. To do this, multiply the first terms, then multiply the first term (first binomial), and the second term (the second binomial), then multiply the second term (the first binomial) and the first term (the second binomial), and then multiply the second terms. For example, consider the product of two binomials (x + 2) (x + 4).

- Multiplying the first terms: (x + 2) (x + 4) = x2 + __
- Multiplying the first term (first binomial) and the second term (the second binomial): (x + 2) (x + 4) = x2 + 4x + __
- Multiplying the second term (the first binomial) and the first term (the second binomial): (x + 2) (x + 4) = x2 + 4x + 2x + __
- Multiplying the second terms: (x + 2) (x + 4) = x2 + 4x + 2x + 8
- Simplification: x2 + 4x + 2x + 8 = x2 + 6x + 8.

Now let’s do the factoring. When multiplying two binomials you get a trinomial of the form ax^{2} + bx + c, where a, b, c are constant coefficients (i.e. numbers). Therefore, we can do the inverse operation – factor the trinomial to the product of two binomials.

- If trinomial is given in another form, move its terms in the desired order. For example, rewrite 3x - 10 + x
^{2}to the following expression: x^{2}+ 3x - 10. - Since this trinomial’s highest exponent is 2 (x
^{2}), then such trinomial is called a square trinomial.

Now, write this trinomial, put an equal sign, and then write the answer in the form of (__ __) (__ __). You will fill in the brackets while doing the factoring. Do not write «+» or «-» between the spaces as the correct signs will be determined in the process of trinomial factoring.

Fill in the gaps in the first two brackets. The simple trinomial, which first term is x^{2}, the first members of both binomial will be x and x, since x * x = x^{2}.

In our example, x^{2} + 3x - 10, the first term - is x^{2}, so write the following: (x __) (x __).

If you return to the first step of, you will see that when multiplying the second terms of binomials, we obtain the free term of the trinomial (the term without a variable «x»). Thus, it is necessary to find two numbers, which when multiplied together will give the free term. In our example, x^{2} + 3x – 10, the free term is equal to -10. Think about what two numbers when multiplied result in -10? These are: -1 * 10; 1 * -10; -2 * 5; 2 * -5.

Now let’s fill the gaps in both parentheses. In the previous step, you received a pair of multipliers (of the free member). Put them in your answer and check whether they comply with the second term of your trinomial:

- In our example, x
^{2}+ 3x - 10, the second term is equal to 3x. - Substitute -1 and 10: (x - 1) (x + 10) = x
^{2}+ 9x - 10. ≠ 3x 9x. Not suitable. - Substitute 1 and -10: (x + 1) (x - 10) = x
^{2}- 9x - 10. -9x≠ 3x. Not suitable. - Substitute -2 and 5: (x - 2) (x + 5) = x
^{2}+ 3x - 10. 3x = 3x. Good. Thus, the correct answer is: (x - 2) (x + 5).

In simple cases, when the variable x^{2} doesn’t have a coefficient, you can do the following: just add the two multipliers (which, when multiplied together result in the free term), and then attribute «x» to the result." In our example (-2 + 5) x = 3x. It will not work during the factoring of complex trinomials, so remember the detailed method described above.

Simplify the complex trinomial to the simple one if it is possible. For example, consider a complex trinomial 3x^{2} + 9x - 30. Determine whether you can put out of the brackets the common factor (which is equal to the greatest common divisor of each member of the trinomial). In our example, we can put 3 out of the brackets:

- 3x
^{2}= (3) (x^{2}) - 9x = (3) (3x)
- -30 = (3) (- 10)

Thus, the expression 3x^{2} + 9x - 30 = (3) (x^{2} + 3x-10). You can factor this simple trinomial as described in the previous section. You will receive: (3) (x-2) (x + 5).

You can also make a more complex simplification. Perhaps, you will need to put out of the brackets the factor with variable or factor the process of putting out of the brackets several times to get a simple trinomial. Here are a few examples:

- 2x
^{2}y + 14xy + 24y = (2y) (x^{2}+ 7x + 12) - x
^{4}+ 11x^{3}- 26x^{2}= (x^{2}) (x2 + 11x - 26) - -x
^{2}+ 6x - 9 = (-1) (x^{2}- 6x + 9)

Do not forget to factor simple trinomial using the method described above.

Factoring trinomials with the coefficient near the x^{2}. Some complex quadratic polynomials can’t be simplified to simple trinomials. For example, factor 3x^{2} + 10x + 8.

- Write the answer in the form of: (__ __) (__ __).
- Fill in the first gaps in two parentheses. Since the 3 * x = 3x
^{2}, you can write the interim response in the form of: (3x __) (x __). - Write couples of factors of the free term of 8: 1 * 8; 2 * 4.
- Fill in the second gaps in both parentheses. Put in response pairs of free terms and check whether they comply with the second member (10x) of your trinomial. Keep in mind that here the order of the factors is important, since the first term of the first binomial equals to 3x, and not just the «x».
- (3x + 1) (x + 8) = 3x
^{2}+ 25x + 8; ≠ 25x 10x. It is not suitable. - (3x + 8) (x + 1) = 3x
^{2}+ 11x + 8; ≠ 11x 10x. It is not suitable. - (3x + 2) (x + 4) = 3x
^{2}+ 14x + 8; ≠ 14x 10x. It is not suitable. - (3x + 4) (x + 2) = 3x
^{2}+ 10x + 8; 10x = 10x. It is good

Use a replacement for the factoring of polynomials of higher degrees, for instance, with term that is equal to x^{4}. Use the bench to lead such a polynomial to a simple polynomial.

You’ve found out how to factor the trinomial using the factoring trinomials calculator and simple formulas.

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