to view answer.When expressing a mathematical expression or equation as the result of a problem, it is important to display it in the simplest form possible. This makes the result easier to read, easier to understand, and is the accepted convention in mathematics. An expression or equation is in its simplest form when all of the operations within the expression are completed using the order of operations and when all like terms have been combined [combine like terms]. The final simplified expression is equivalent to the original expression.
Simplify the following expression:
$${\Large\frac{8x+3\left(5x+x^2-2\right)}{4+6 \cdot 2}}$$
1. Following the order of operations, the first thing we must do is simplify the expresson inside the parentheses. It is important to note that the division bar serves as a grouping symbol which groups all terms of the numerator together, as well as all the terms of the denominator. Thus, it is equivalent to place parentheses around the numerator and around the denominator:
\[{\Large\frac{8x+3\left(5x+x^2-2\right)}{4+6 \cdot 2}}={\Large\frac{\left[8x+3\left(5x+x^2-2\right)\right]}{\left[4+6 \cdot 2\right]}}\]
2. At this point, we may either simplify the numerator or the denominator. It does not matter which is simplified first, but we must simplify everything in each set of parentheses before performing division. We will begin by simplifying the expression in the denominator.
Following the order of operations, we multiply first:
\[{\frac{\left[8x+3\left(5x+x^2-2\right)\right]}{\left[4+6 \cdot 2\right]}}= {\frac{\left[8x+3\left(5x+x^2-2\right)\right]}{\left[4+12\right]}}\]
And then, we add:
\[{\frac{\left[8x+3\left(5x+x^2-2\right)\right]}{\left[4+12\right]}}={ \frac{\left[8x+3\left(5x+x^2-2\right)\right]}{16}}\]
3. Next, we need to simplify the expression in the numerator. Notice that inside this top set of parentheses is another set of parentheses. According to the order of operations, we must always start with the innermost parentheses. So, first we must simplify:
\[\left(5x+x^2-2\right)\]
Here, check to see if there are any terms that can be combined or operations that can be performed. Since the x’s have different exponents, there are no like terms, and there are also no operations that can be performed. Thus, inside the parentheses the expression is in its simplest form and cannot be simplified further.
4. Now, following the order of operations again, we multiply the 3 by the expression in parentheses. This will require use of the distributive property. Distributing the 3 to each of the terms in parentheses, we have:
\[{\frac{\left[8x+35x+x^2-2\right]}{16}}={\frac{\left[8x+\left(15x+3x^2-6\right)\right]}{16}}\]
Note:
Make sure to pay attention to negatives. It may help to rewrite $$3 \cdot \left(5x+x^2-2\right)$$ as $$3 \cdot \left(5x+x^2+(-2)\right)$$. Remember that subtracting is the same as adding a negative.
5. We can now combine like terms in the numerator. Since we have two terms, 8x and 15x, that have the same variable with the same exponent, these can be combined as:
$${\Large\frac{\left[8x+15x+3x^2-6\right]}{16}}= {\Large\frac{\left[23x+3x^2-6\right]}{16}}$$
6. To simplify further, the division by 16 can be performed on the expression in the numerator. When dividing an expression by a number, like here, we need to divide each term in the numerator by 16:
$${\Large\frac{23x+3x^2-6 \:}{16}}={\Large\frac{23x}{16}+\frac{3x^2}{16}}-{\Large\frac{6}{16}}$$
Note: To understand why this works, think about a completely numerical case, like:
$${\Large\frac{1+1+2}{4}}={\Large\frac{4}{4}}=1$$
or
$${\Large\frac{1+1+2}{4}}={\Large\frac{1}{4}}+{\Large\frac{1}{4}}+{\Large\frac{2}{4}}=1$$
7. The last thing that can be done to simplify this expression is to simplify fractions. The fractions $${\Large\frac{23}{16}}$$ and $${\Large\frac{3}{16}}$$ are in simplest form, but $${\Large\frac{6}{16}}$$ can be simplified to $${\Large\frac{3}{8}}$$ since both numbers are divisible by 2. Therefore, the final simplified result is:
\[{\frac{23}{16}}x+{\frac{3}{16}}x^2-{\frac{3}{8}}\]
8. Thus the expression $${\Large\frac{8x+3\left(5x+x^2-2\right)}{4+6 \cdot 2}}$$ is equivalent to $${\Large\frac{23}{16}}x+{\Large\frac{3}{16}}x^2-{\Large\frac{3}{8}}$$ The final answer could be written in any of the following ways:
$${\Large\frac{23}{16}}x+{\Large\frac{3}{16}}x^2-{\Large\frac{3}{8}}$$
$${\Large\frac{23}{16}}x-{\Large\frac{3}{8}}+{\Large\frac{3}{16}}x^2$$
$$-{\Large\frac{3}{8}}+{\Large\frac{23}{16}}x+{\Large\frac{3}{16}}x^2$$
$$-{\Large\frac{3}{8}}+{\Large\frac{3}{16}}x^2+{\Large\frac{23}{16}}x $$
$${\Large\frac{3}{16}}x^2-{\Large\frac{3}{8}}+{\Large\frac{23}{16}}x $$
$${\Large\frac{3}{16}}x^2+{\Large\frac{23}{16}}x -{\Large\frac{3}{8}} $$
Note: There are times when it is convenient to convert fractions to decimals, but in general, it is best to leave mathematical expressions and solutions in their fraction form. The reason for this is that some fractions must be rounded to write them in decimal form, making the answer less accurate.
The last form on this list is the most common mathematical way to write this type of expression. In this form, the exponents are in descending order. This means that they are listed from the highest power of x, which is 2, to the lowest power of x, which is 0.
Some practice problems to check your skills:
Simplify:
1. $$3x + 8x^2 - 2x + 2x^3 - 6\left( {3 + x} \right)$$
Think about it for a moment and then access this link to view answer.
2. $${\Large{{6y + 3\left( {4y - 8y^2 } \right)} \over {2 \cdot 3 - 4}}}$$
Think about it for a moment and then access this link to view answer.
3. $${\Large{{2 \cdot 3y + 7x - 2 \cdot 6} \over {4 + 5}}}$$
Think about it for a moment and then access this link to view answer.
4. $$3\left( {4x + 2} \right) + {\Large{{{2\left( {3x - 5x^2 } \right)} \over {\left( {{{12} \over 4}} \right)}}}}$$
Think about it for a moment and then access this link to view answer.
5. $${\Large{{4 \cdot \left[ {2\left( {3x + 4x^2 + 4} \right) + 3x} \right]} \over {3\left( {7 - 3} \right) - 2 \cdot 2}}}$$
Think about it for a moment and then access this link to view answer.