Simplifying Algebraic Expressions with Parentheses

Hello! Today we’re going to take a look at simplifying algebraic expressions with parentheses. Before we do that, let’s review what like terms are. Like terms are any two terms that have the same variable raised to the same exponent. Some examples are:

\(x^{2}\) and \(7x^{2}\)
 
\(3xy\) and \(4xy\)
 
\(2x\) and \(5x\)

 

Now, an example of two terms that are not like terms would be:

\(6x\) and \(6x^{2}\)

 

Even though they have the same coefficient and the same variable, the variables are not raised to the same power, so these two terms are not like terms.

The coefficient for like terms can be different; in fact, they usually are. For example, \(2x\) and \(5x\) are like terms, even though the coefficient 2 is different from the coefficient 5.

Now that we’ve reviewed what like terms are, let’s use this information to simplify some expressions. We’re going to start with this expression:

\((3x^{2}+2x-5)+(7x^{2}-3x+14)\)

 

Since all our terms are being added or subtracted, we can use the associative and commutative properties to regroup our terms so our like terms are next to one another. Remember, subtracting can be thought of as adding a negative, so these properties will still work. So, let’s combine our like terms. Our like terms are \(3x^{2}\) and \(7x^{2}\), \(2x\) and \(-3x\), and \(-5\) and \(14\). So we’re going to regroup and write them this way:

\((3^{2}+7x)+(2x-3x)+(-5+14)\)

 

Now we can simplify each set of parentheses.

\(10x^{2}+(-x)+9\)

 

Now we can simplify this just a little bit further by changing this “adding a negative number” to just subtracting. So our final answer is going to be:

\(10x^{2}-x+9\)

 

Since this isn’t an equation, we can’t solve for anything. The most we can do is simplify, so this is our final answer. Let’s try another example.

\((x-17x)+(2xy-9xy)+(-3y+27y)\)

 

First, we regroup our terms so our like terms are together. So let’s start by underlining our like terms. We have \(x\) and \(-17x\) as like terms, \(2xy\) and \(-9xy\) are like terms, and \(-3y\) and \(27y\) are also like terms. So now we can write them in parentheses.

\((x-17x)+(2xy-9xy)+(-3y+27y)\)

 

And now all we have to do is simplify each set of parentheses.

\(-16x-7xy+24y\)

 

Before we go, I want to try one more example that’s slightly more difficult.

\((6x^{2}+4x-2)-(2x^{2}-18x+5)\)

 

Notice that this expression has a minus sign between the two sets of parentheses. Remember how I said earlier that subtracting is the same thing as adding a negative? This will help us in simplifying this expression. Let’s rewrite it like this:

\((6x^{2}+4x-2)+(-1)(2x^{2}+18x-5)\)

 

This is where the adding a negative number comes in. But the negative number has to be 1 because that doesn’t change the value of the actual expression; 1 times anything is the original expression. So, this doesn’t change the value of anything, it just makes it look slightly different so we can understand where our next steps come from.

Now that we have a negative 1 here, we need to distribute this negative 1 into each term in the second set of parentheses. So, we’re going to leave this first set of parentheses exactly the same, and distribute the -1 to the second set.

\((6x^{2}+4x-2)+(-2x^{2}+18x-5)\)

 

Notice that we have the exact same set of parentheses, just the sign has changed in front of each term. Now we’re going to follow the exact same steps as we have before. We’re going to underline our like terms, so we have \(6x^{2}\) and \(-2x^{2}\). \(4x\) and \(18x\) are also like terms, and so are -2 and -5. So now we’re going to regroup these like terms into their own sets of parentheses.

\((6x^{2}-2x^{2})+(4x-18x)+(-2-5)\)

 

Now we’ll simplify each set of parentheses.

\(4x^{2}+22x-7\)

 

And there you have it! I hope this video made this topic more understandable for you. Thanks for watching and happy studying!

Practice Questions

Question #1:

 
Completely simplify the expression: \((5x^2-7x+6)+({3x}^2+4x+8)\).

\(9x^2-4x-2\)
\({8x}^2-3x+14\)
\({2x}^2+3x+2\)
\({8x}^2+3x+14\)
Answer:

To simplify an algebraic expression, we want to combine like terms. Like terms are terms whose variables with any exponents are the same.
First, regroup the terms by using the associative and commutative properties of addition so our like terms are together.

\((5x^2+3x^2)+(-7x+4x)+(6+8)\)

Now, simplify each group of like terms by combing them.

\(8x^2+(-3x)+14\)

Lastly, \(+(-3x)\) simplifies to \(-3x\).

\(8x^2-3x+14\)

There are no more like terms to combine, the expression is completely simplified.

Question #2:

 
Completely simplify the expression: \((16x-9xy+13y)+(-12x+6xy-8y)\).

\(4x-3xy+5y\)
\(-4x+3xy+5y\)
\(28x-3xy-5y\)
\(22x-18xy-5y\)
Answer:

To simplify an algebraic expression, we want to combine like terms. Like terms are terms whose variables with any exponents are the same.
First, regroup the terms by using the associative and commutative properties of addition so our like terms are together.

\((16x-12x)+(-9xy+6xy)+(13y-8y)\)

Now, simplify each group of like terms by combing them.

\(4x+(-3xy)+5y\)

Lastly, simplify \(+(-3xy)\) to \(-3xy\).

\(4x-3xy+5y\)

Since there are no more like terms to combine, the expression is completely simplified.

Question #3:

 
Completely simplify the expression: \((6x^2+2x+7)-({2x}^2-5x+13)\).

\({8x}^2+7x-6\)
\({8x}^2-3x+20\)
\({4x}^2+7x-6\)
\({4x}^2+3x+20\)
Answer:

To simplify an algebraic expression, we want to combine like terms. Like terms are terms whose variables with any exponents are the same.
Before combining like terms, we can change the subtraction sign between the two expressions to an addition sign by “adding the negative of it”.

\((6x^2+2x+7)+(-1)({2x}^2-5x+13)\)

The –1 gets distributed to the second expression to give us:

\((6x^2+2x+7)+(-{2x}^2+5x-13)\)

Now, regroup the terms by using the associative and commutative properties of addition so our like terms are together.

\((6x^2-2x^2)+(2x+5x)+(7-13)\)

Simplify each group of like terms by combing them.

\(4x^2+7x+(-6)\)

Lastly, simplify \(+(-6)\) to \(-6\).

\(4x^2+7x-6\)

There are no more like terms to combine, the expression is completely simplified.

Question #4:

 
Suppose the area, in square feet, of a square can be represented by the algebraic expression \(16x^2+48x+36\), and the area, in square feet, of a rectangle can be represented by the algebraic expression \(5x^2-54x+40\). Which of the following algebraic expressions represents the combined area of the square and rectangle?

\({53x}^2-38x+76\)
\(-38x^2+53x+76\)
\(21x^2+6x+76\)
\(21x^2-6x+76\)
Answer:

To find the combined area of the square and the rectangle, we need to add the two expressions that represent their respective areas.

\((16x^2+48x+36)+(5x^2-54x+40)\)

To simplify the algebraic expression that represents the combined area, combine like terms.

First, regroup the terms by using the associative and commutative properties of addition so our like terms are together.

\((16x^2+5x^2)+(48x-54x)+(36+40)\)

Now, simplify each group of like terms by combing them.

\(21x^2+(-6x)+76\)

Lastly, simplify adding \(+(-6x)\) to \(-6x\).

\(21x^2-6x+76\)

Since there are no more like terms to combine, the expression for the combined area is completely simplified, so the combined area is \(21x^2-6x+76\) square feet.

Question #5:

 
On the first day of a sale, the profit generated, in dollars, of selling a new brand of an exterior paint is represented by the algebraic expression \(2x+5xy+9y\). On the second day of the sale, the profit generated for selling the same paint is represented by the algebraic expression \(x+7xy-2y\). Which of the following algebraic expressions represents is the difference in profit from the first day of the sale to the second day?

\(3x-12xy-7y\)
\(x+2xy+11y\)
\(x-2xy+11y\)
\(3x-2xy+11y\)
Answer:

To find the difference in profit from the first and second day of the sale, we must take the difference in the two expressions representing the profits from the respective days.

\((2x+5xy+9y)-(x+7xy-2y)\)

To simplify an algebraic expression, we want to combine like terms.

Before combining like terms, we can change the subtraction sign between the two expressions to an addition sign by “adding the negative of it”.

\((2x+5xy+9y)+(-1)(x+7xy-2y)\)

The –1 gets distributed to the second expression to give us:

\((2x+5xy+9y)+(-x-7xy+2y)\)

Now, regroup the terms by using the associative and commutative properties of addition so our like terms are together.

\((2x-x)+(5xy-7xy)+(9y+2y)\)

Simplify each group of like terms by combing them.

\(x+(-2xy)+11y\)

Lastly, simplify \(+(-2xy)\) to \(-2xy\).

\(x-2xy+11y\)

There are no more like terms to combine, the expression is completely simplified, so the difference in the profit from the first day to the second day of the sale is \(x-2xy+11y\) dollars.

 

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by Mometrix Test Preparation | This Page Last Updated: May 13, 2022