# The Distributive Law and Combining Like Terms

## The distributive law

In each of the rows of this table, do both arithmetic problems. The problems will be illustrated on the grids to the left.

As you can see, both answers are the same in each row of the table. This illustrates the following rule:

The distributive law of multiplication over addition: for any three numbers \$a\$, \$b\$, and \$c\$,
\$\$ \cl"tight"{\table a(b+c), =, a(b)+a(c) ; (b+c)(a), =, b(a)+c(a)} \$\$

Do both arithmetic problems in each row.

Again, both answers in each row are the same, illustrating:

The distributive law of multiplication over subtraction: for any three numbers \$a\$, \$b\$, and \$c\$,
\$\$ \cl"tight"{\table a(b-c), =, a(b)-a(c) ; (b-c)(a), =, b(a)-c(a)} \$\$

## Expanding \$a(bx+c)\$

In each row of the table below, enter the values of the expressions \$3(x+2)\$, \$3x+3(2)\$, and \$3x+6\$ for the given value of \$x\$, as illustrated on the grids to the left.

\$x\$\$\cl"tight"{\table \colspan 3 3(x+2) ; 3x, +, 3(2) ; 3x, +, 6}\$

That is, you can use the distributive law to turn expressions which need parentheses (like \$3(x+2)\$) into expressions which do not (like \$3x+6\$). This is called expanding an expression.

Each row of the table below has some expression that requires parentheses to write. Use the distributive law to expand the expression, as shown in the example in the first row.

Use the distributive law to expand each expression in the table below, and then simplify the result.

Taking the opposite of a number is the same as multiplying that number by \$-1\$. For example, \$-1(3)=-3\$, which is the opposite of \$3\$. This is also true for expressions: the opposite of \$x+3\$ — that is, \$-(x+3)\$ — is the same as \$-1\$ times \$x+3\$ — that is, \$-1(x+3)\$. So you can use the distributive law to expand the opposite of an expression.

In each of the rows of this table, find and expand the opposite of the given expression, as shown in the example row.

In negateExpr, you used the fact that taking opposites is the same as multiplying by \$-1\$, in order to use the distributive law. For example, you would have negated \$2x-3\$ by writing:

\$\$ -(2x-3) = (-1)(2x-3) = (-1)(2x) - (-1)(3) = -2x - (-3) = -2x + 3 \$\$

You can make this algebra simpler by directly distributing the first \$-\$ sign to each term in the expression. For example, you can write:

\$\$-(2x-3) = -(2x) - (-3) = (-2x) - (-3) = -2x+3\$\$

Find and expand the opposite of the expressions in the table below, as shown in the example row.

## Combining like terms

In each row of the table below, enter the values of the expressions \$4x+5x\$, \$(4+5)x\$, and \$9x\$ for the given value of \$x\$, as illustrated on the grids to the left.

\$x\$\$\cl"tight"{\table 4x+5x ; (4+5)x ; 9x}\$

You can use the distributive law in this way to add any two multiples of \$x\$. For example, you just simplified \$4x+5x\$ into \$9x\$. This is called combining like terms.

Each row of the table below has some expression in which like terms are added or subtracted. Combine those like terms, as shown in the example in the first row.

Combine the like terms in each of these expressions, remembering that \$x\$ is the same as \$1x\$, and \$-x\$ is the same as \$-1x\$.