Content

• Steps
• Part 1 of 3: Factoring binomials
• Part 2 of 3: Factoring binomials for solving equations
• Part 3 of 3: Solving Complex Problems
• Tips
• Warnings

A binomial (binomial) is a mathematical expression with two terms between which there is a plus or minus sign, for example, ${ displaystyle ax + b}$... The first member includes the variable, and the second includes or does not include it. Factoring a binomial involves finding terms that, when multiplied, produce the original binomial in order to solve or simplify it.

Steps

Part 1 of 3: Factoring binomials

1. 1 Understand the basics of the factoring process. When factoring a binomial, the factor that is a divisor of each term of the original binomial is taken out of the bracket. For example, the number 6 is completely divisible by 1, 2, 3, 6. Thus, the divisors of the number 6 are the numbers 1, 2, 3, 6.
   • Divisors 32: 1, 2, 4, 8, 16, 32.
   • The divisors of any number are 1 and the number itself. For example, divisors of 3 are 1 and 3.
   • Integer divisors can only be integers. The number 32 can be divided by 3.564 or 21.4952, but you get not an integer, but a decimal fraction.
2. 2 Order the terms of the binomial to facilitate the factoring process. A binomial is the sum or difference of two terms, at least one of which contains a variable. Sometimes variables are raised to a power, for example, ${ displaystyle x ^ {2}}$ or ${ displaystyle 5y ^ {4}}$... It is better to order the terms of the binomial in ascending order of exponents, that is, the term with the smallest exponent is written first, and with the largest - the last. For example:
   • ${ displaystyle 3t + 6}$ → ${ displaystyle 6 + 3t}$
   • ${ displaystyle 3x ^ {4} + 9x ^ {2}}$ → ${ displaystyle 9x ^ {2} + 3x ^ {4}}$
   • ${ displaystyle x ^ {2} -2}$ → ${ displaystyle -2 + x ^ {2}}$
     • Notice the minus sign in front of 2. If a term is subtracted, write a minus sign in front of it.
3. 3 Find the greatest common divisor (GCD) of both terms. GCD is the largest number by which both members of the binomial are divisible. To do this, find the divisors of each term in the binomial, and then select the greatest common divisor. For example:
   • A task:${ displaystyle 3t + 6}$.
     • Divisors 3: 1, 3
     • Divisors 6: 1, 2, 3, 6.
     • GCD = 3.
4. 4 Divide each term in the binomial by the Greatest Common Divisor (GCD). Do this to factor out the GCD. Note that each member of the binomial decreases (because it is divisible), but if the GCD is excluded from the parenthesis, the final expression will be equal to the original one.
   • A task:${ displaystyle 3t + 6}$.
   • Find the GCD: 3
   • Divide each binomial term by gcd:${ displaystyle { frac {3t} {3}} + { frac {6} {3}} = t + 2}$
5. 5 Move the divisor out of the parentheses. Earlier, you divided both terms of the binomial by the divisor 3 and got ${ displaystyle t + 2}$... But you cannot get rid of 3 - in order for the values ​​of the initial and final expressions to be equal, you need to put 3 outside the parentheses, and write the expression obtained as a result of division in parentheses. For example:
   • A task:${ displaystyle 3t + 6}$.
   • Find the GCD: 3
   • Divide each binomial term by gcd:${ displaystyle { frac {3t} {3}} + { frac {6} {3}} = t + 2}$
   • Multiply the divisor by the resulting expression:${ displaystyle 3 (t + 2)}$
   • Answer: ${ displaystyle 3 (t + 2)}$
6. 6 Check your answer. To do this, multiply the term before the brackets by each term inside the brackets. If you get the original binomial, the solution is correct. Now solve the problem ${ displaystyle 12t + 18}$:
   • Order the members:${ displaystyle 18 + 12t}$
   • Find the GCD:${ displaystyle 6}$
   • Divide each binomial term by gcd:${ displaystyle { frac {18t} {6}} + { frac {12t} {6}} = 3 + 2t}$
   • Multiply the divisor by the resulting expression:${ displaystyle 6 (3 + 2t)}$
   • Check the answer:${ displaystyle (6 * 3) + (6 * 2t) = 18 + 12t}$

Part 2 of 3: Factoring binomials for solving equations

1. 1 Factor the binomial to simplify it and solve the equation. At first glance, it seems impossible to solve some equations (especially with complex binomials). For example, solve the equation ${ displaystyle 5y-2y ^ {2} = - 3y}$... There are powers in this equation, so factor the expression first.
   • A task:${ displaystyle 5y-2y ^ {2} = - 3y}$
   • Remember that a binomial has two members. If the expression includes more terms, learn how to solve polynomials.
2. 2 Add or subtract some monomial to both sides of the equation so that zero remains on one side of the equation. In the case of factorization, the solution to equations is based on the immutable fact that any expression multiplied by zero is equal to zero. Therefore, if we equate the equation to zero, then any of its factors must be equal to zero. Set one side of the equation to 0.
   • A task:${ displaystyle 5y-2y ^ {2} = - 3y}$
   • Set to zero:${ displaystyle 5y-2y ^ {2} + 3y = -3y + 3y}$
     • ${ displaystyle 8y-2y ^ {2} = 0}$
3. 3 Factor the resulting bin. Do this as described in the previous section. Find the greatest common factor (GCD), divide both terms of the binomial by it, and then move the factor out of the parentheses.
   • A task:${ displaystyle 5y-2y ^ {2} = - 3y}$
   • Set to zero:${ displaystyle 8y-2y ^ {2} = 0}$
   • Factor:${ displaystyle 2y (4-y) = 0}$
4. 4 Set each factor to zero. In the resulting expression, 2y is multiplied by 4 - y, and this product is zero. Since any expression (or term) multiplied by zero is zero, then 2y or 4 - y is 0. Set the resulting monomial and binomial to zero to find "y".
   • A task:${ displaystyle 5y-2y ^ {2} = - 3y}$
   • Set to zero:${ displaystyle 8y-2y ^ {2} + 3y = 0}$
   • Factor:${ displaystyle 2y (4-y) = 0}$
   • Set both factors to 0:
     • ${ displaystyle 2y = 0}$
     • ${ displaystyle 4-y = 0}$
5. 5 Solve the resulting equations to find the final answer (or answers). Since each factor equates to zero, the equation can have multiple solutions. In our example:
   • ${ displaystyle 2y = 0}$
     • ${ displaystyle { frac {2y} {2}} = { frac {0} {2}}}$
     • y = 0
   • ${ displaystyle 4-y = 0}$
     • ${ displaystyle 4-y + y = 0 + y}$
     • y = 4
6. 6 Check your answer. To do this, substitute the found values ​​into the original equation. If the equality is true, then the decision is correct. Substitute the found values ​​instead of "y". In our example, y = 0 and y = 4:
   • ${ displaystyle 5 (0) -2 (0) ^ {2} = - 3 (0)}$
     • ${ displaystyle 0 + 0 = 0}$
     • ${ displaystyle 0 = 0}$This is the right decision
   • ${ displaystyle 5 (4) -2 (4) ^ {2} = - 3 (4)}$
     • ${ displaystyle 20-32 = -12}$
     • ${ displaystyle -12 = -12}$And this is the right decision

Part 3 of 3: Solving Complex Problems

1. 1 Remember that a term with a variable can also be factorized, even if the variable is raised to a power. When factoring, you need to find a monomial that divides each member of the binomial integrally. For example, the monomial ${ displaystyle x ^ {4}}$ can be factorized ${ displaystyle x * x * x * x}$... That is, if the second term of the binomial also contains the variable "x", then "x" can be taken out of the brackets. Thus, treat variables as integers. For example:
   • Both members of the binomial ${ displaystyle 2t + t ^ {2}}$ contain "t", so "t" can be taken out of the parenthesis: ${ displaystyle t (2 + t)}$
   • Also, a variable raised to a power can be taken out of the bracket. For example, both members of the binomial ${ displaystyle x ^ {2} + x ^ {4}}$ contain ${ displaystyle x ^ {2}}$, so ${ displaystyle x ^ {2}}$ can be taken out of the bracket: ${ displaystyle x ^ {2} (1 + x ^ {2})}$
2. 2 Add or subtract similar terms to get a binomial. For example, given the expression ${ displaystyle 6 + 2x + 14 + 3x}$... At first glance, this is a polynomial, but in fact, this expression can be converted to a binomial. Add similar terms: 6 and 14 (do not contain a variable), and 2x and 3x (contain the same variable "x"). In this case, the process of factoring will be simplified:
   • Original expression:${ displaystyle 6 + 2x + 14 + 3x}$
   • Order the members:${ displaystyle 2x + 3x + 14 + 6}$
   • Add similar terms:${ displaystyle 5x + 20}$
   • Find the GCD:${ displaystyle 5 (x) +5 (4)}$
   • Factor:${ displaystyle 5 (x + 4)}$
3. 3 Factor the difference of perfect squares. A perfect square is a number whose square root is an integer, for example ${ displaystyle 9}$${ displaystyle (3 * 3)}$, ${ displaystyle x ^ {2}}$${ displaystyle (x * x)}$ and even ${ displaystyle 144t ^ {2}}$${ displaystyle (12t * 12t)}$... If the binomial is the difference of perfect squares, for example, ${ displaystyle a ^ {2} -b ^ {2}}$, then it is factorized by the formula:
   • Difference of squares formula:${ displaystyle a ^ {2} -b ^ {2} = (a + b) (a-b)}$
   • A task:${ displaystyle 4x ^ {2} -9}$
   • Extract the square roots:
     • ${ displaystyle { sqrt {4x ^ {2}}} = 2x}$
     • ${ displaystyle { sqrt {9}} = 3}$
   • Substitute the found values ​​into the formula: ${ displaystyle 4x ^ {2} -9 = (2x + 3) (2x-3)}$
4. 4 Factor the difference between the complete cubes. If the binomial is the difference of complete cubes, for example, ${ displaystyle a ^ {3} -b ^ {3}}$, then it is factorized using a special formula. In this case, it is necessary to extract the cube root from each member of the binomial, and substitute the found values ​​into the formula.
   • The formula for the difference between cubes:${ displaystyle a ^ {3} -b ^ {3} = (a-b) (a ^ {2} + ab + b ^ {2})}$
   • A task:${ displaystyle 8x ^ {3} -27}$
   • Extract cubic roots:
     • ${ displaystyle { sqrt [{3}] {8x ^ {3}}} = 2x}$
     • ${ displaystyle { sqrt [{3}] {27}} = 3}$
   • Substitute the found values ​​into the formula: ${ displaystyle 8x ^ {3} -27 = (2x-3) (4x ^ {2} + 6x + 9)}$
5. 5 Factor the sum of the full cubes. Unlike the sum of perfect squares, the sum of complete cubes, for example, ${ displaystyle a ^ {3} + b ^ {3}}$, can be factorized using a special formula. It is similar to the formula for the difference between cubes, but the signs are reversed. The formula is quite simple - to use it, find the sum of full cubes in the problem.
   • The formula for the sum of cubes:${ displaystyle a ^ {3} + b ^ {3} = (a + b) (a ^ {2} -ab + b ^ {2})}$
   • A task:${ displaystyle 8x ^ {3} -27}$
   • Extract cubic roots:
     • ${ displaystyle { sqrt [{3}] {8x ^ {3}}} = 2x}$
     • ${ displaystyle { sqrt [{3}] {27}} = 3}$
   • Substitute the found values ​​into the formula: ${ displaystyle 8x ^ {3} -27 = (2x + 3) (4x ^ {2} -6x + 9)}$

Tips

• Sometimes binomial members do not have a common divisor. In some tasks, members are presented in a simplified form.
• If you can't find GCD right away, start by dividing by small numbers. For example, if you don't see that the GCD of numbers 32 and 16 is 16, divide both numbers by 2. You get 16 and 8; these numbers can be divided by 8. Now you get 2 and 1; these numbers cannot be reduced. Thus, it is obvious that there is a larger number (compared to 8 and 2), which is the common divisor of the two given numbers.
• Note that sixth-order terms (with an exponent of 6, for example x) are both perfect squares and perfect cubes. Thus, to binomials with terms of the sixth order, for example, x - 64, one can apply (in any order) the formulas for the difference of squares and the difference of cubes. But it is better to first apply the formula for the difference of squares in order to more correctly decompose with a binomial.

Warnings

• A binomial, which is the sum of perfect squares, cannot be factorized.