# Perform the indicated operation and simplify your answer: x^{2} + 5 x + 6 x^{2} + 7 x + 12 x^{2} + 5 x + 4 x^{2} + 3 x + 2

Algebraic expression started in the 9th century. In the beginning, it was more in statement form and not mathematical at all. For instance, algebraic equations used to be written as “5 times the thing added with 3 gives 18” which is basically 5x + 3 = 18. This type of equation which was not mathematical was Babylonian algebra. Algebra evolved with time and with the different forms provided. It started with Egyptian algebra, then came Babylonian algebra, then came greek geometrical algebra, moved to diophantine algebra, followed by Hindu algebra, then came Arabic algebra, and followed by abstract algebra. Today, the easiest and most convenient form of algebra is taught in classes for better understanding.

### Algebraic Expressions

Algebraic expressions are the expressions obtained from the combination of variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and so on. An algebraic expression is made up of terms, there can be one or more than one term present in the equation. Let’s learn about the basic terms used in algebraic expressions,

### Constants, Variables, Coefficients, and Terms

In the algebraic expression, fixed numerical are called constants, constants do not have any variables attached to them. For example, 3x – 1 has a constant -1 to it. Variables are the unknown values that are present in the algebraic expression, for instance, 4y + 5z has y and z as variables. Coefficients are the fixed values (real numbers) attached to the variables, they are multiplied with the variables. For example, 5x^{2} + 3 has 5 as the coefficient of x^{2}. A term can be a constant, a variable, or a combination of both, basically, each term is separated by either addition or subtraction. For example, 3x + 5, 3x and 5 are the terms.

### Simplifying Algebraic expressions

Simplifying algebraic expressions is easy and very basic. First, understand what are like and unlike terms, like terms have the same sign and unlike terms have opposite signs. In order to simplify the given algebraic expression, first, find out the terms having the same power, then if the terms are like terms, then add them, if they are unlike terms, find the difference of the terms. The most simplified form of an algebraic expression is the one where no same power terms are not repeated.

For instance, lets simplify 4x^{5 }+ 3x^{3} – 8x^{2} + 67 – 4x^{2} + 6x^{3}, the same powers that are repeated are cubic and square, upon combining them together, the expression becomes, 4x^{5} + (3x^{3} + 6x^{3}) – (8x^{2} – 4x^{2}) + 67. Now, simplifying the expression, the final answer obtained is, 4x^{5} + 9x^{3} – 12x^{2} + 67. This term does not have any terms repeated that have the same power.

### Perform the indicated operation and simplify your answer: x^{2} + 5x + 6x^{2} + 7x + 12x^{2} + 5x + 4x^{2} + 3x + 2.

**Solution:**

The algebraic term mentioned in the problem statement “x

^{2}+ 5x + 6x^{2}+ 7x + 12x^{2}+ 5x + 4x^{2}+ 3x + 2″ has 9 terms. However, these terms can solved and simplified further. Lets take a look at the following steps,

Step 1:Collect the terms having similar powers together(x

^{2}+ 6x^{2}+ 12x^{2}+ 4x^{2}) + (5x + 7x + 5x + 3x) + 2

Step 2:Simplify the terms23x

^{2}+ 20x + 2.

### Sample Problems

**Question 1: Find out the constant from the following algebraic expressions,**

**11x**^{3}+ 3x^{2}– 6**29 + y**^{5}

**Answer:**

Constants are the terms that do not have any variable attached to them, therefore, in the first case, -6 is the constant, and in the second case, 9 is the constant.

**Question 2: Simplify the algebraic term, 2z ^{5} + 11z^{3} – 6y^{6} + 8z^{5} – 2y^{6} + 14 + 7z^{3}**

**Solution:**

In the expression, there are terms with the same power and same variable that are repeated, first bring them together,

(2z

^{5}+ 8z^{5}) + (11z^{3}+ 7z^{3}) – (6y^{6}– 2y^{6}) + 14.Now, simplify the expression,

10z

^{5}+ 18z^{3}– 8y6 + 14.

**Question 3: Simplify the algebraic expression, 4x ^{2 }+ 7x + 12x^{2} + x + 18x^{2} + 8x + 2x^{2} + 9x + 45.**

**Solution:**

The algebraic term mentioned in the problem statement “4x

^{2}+ 7x + 12x^{2}+ x + 18x^{2}+ 8x + 2x^{2}+ 9x + 45″ has 9 terms. However, these terms can solved and simplified further. Lets take a look at the following steps,Step 1: Collect the terms having similar powers together

(4x

^{2}+ 12x^{2}+ 18x^{2}+ 2x^{2}) + (7x + x + 8x + 9x) + 45Step 2: Simplify the terms

36x

^{2}+ 25x + 45.