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$(a+b)(c+d)=ac+ad+bc+bd$
This method is commonly called the FOIL method .
$$(a+b)(2+3)=\underset{2\text{\hspace{0.17em}}\text{terms}}{\underbrace{(a+b)+(a+b)}}+\underset{3\text{\hspace{0.17em}}\text{terms}}{\underbrace{(a+b)+(a+b)+(a+b)}}$$
Rearranging,
$$\begin{array}{l}=a+a+b+b+a+a+a+b+b+b\\ =2a+2b+3a+3b\end{array}$$
Combining like terms,
$$=5a+5b$$
This use of the distributive property suggests the following rule.
Perform the following multiplications and simplify.
With some practice, the second and third terms can be combined mentally.
$$\begin{array}{lll}{(m-3)}^{2}\hfill & =\hfill & (m-3)(m-3)\hfill \\ \hfill & =\hfill & m\cdot m+m(-3)-3\cdot m-3(-3)\hfill \\ \hfill & =\hfill & {m}^{2}-3m-3m+9\hfill \\ \hfill & =\hfill & {m}^{2}-6m+9\hfill \end{array}$$
$$\begin{array}{llll}{(x+5)}^{3}\hfill & =\hfill & (x+5)(x+5)(x+5)\hfill & \text{Associate}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{two}\text{\hspace{0.17em}}\text{factors}\text{.}\hfill \\ \hfill & =\hfill & \left[(x+5)(x+5)\right](x+5)\hfill & \hfill \\ \hfill & =\hfill & \left[{x}^{2}+5x+5x+25\right](x+5)\hfill & \hfill \\ \hfill & =\hfill & \left[{x}^{2}+10x+25\right](x+5)\hfill & \hfill \\ \hfill & =\hfill & {x}^{2}\cdot x+{x}^{2}\cdot 5+10x\cdot x+10x\cdot 5+25\cdot x+25\cdot 5\hfill & \hfill \\ \hfill & =\hfill & {x}^{3}+5{x}^{2}+10{x}^{2}+50x+25x+125\hfill & \hfill \\ \hfill & =\hfill & {x}^{3}+15{x}^{2}+75x+125\hfill & \hfill \end{array}$$
Find the following products and simplify.
$(2{x}^{2}{y}^{3}+x{y}^{2})(5{x}^{3}{y}^{2}+{x}^{2}y)$
$10{x}^{5}y{}^{5}+7{x}^{4}{y}^{4}+{x}^{3}{y}^{3}$
Perform the following additions and subtractions.
$$\begin{array}{ll}3x+7+(x-3).\hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{parentheses}\text{.}\text{\hspace{0.17em}}\text{They}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{preceded}\text{\hspace{0.17em}}\text{by}\hfill \\ \hfill & \text{a}\text{\hspace{0.17em}}"+"\text{\hspace{0.17em}}\text{sign,}\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{them}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{leave}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{}\text{each}\hfill \\ \hfill & \text{term}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{.}\hfill \\ 3x+7+x-3\hfill & \text{Combine}\text{\hspace{0.17em}}\text{like}\text{\hspace{0.17em}}\text{terms}\text{.}\hfill \\ 4x+4\hfill & \hfill \end{array}\text{\hspace{0.17em}}$$
$\begin{array}{ll}5{y}^{3}+11-(12{y}^{3}-2).\hfill & \text{We}\text{\hspace{0.17em}}\text{first}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{parentheses}\text{.}\text{\hspace{0.17em}}\text{They}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{preceded}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{a}\hfill \\ \hfill & \text{"-"}\text{\hspace{0.17em}}\text{sign,}\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{remove}\text{\hspace{0.17em}}\text{them}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{change}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{sign}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{each}\hfill \\ \hfill & \text{term}\text{\hspace{0.17em}}\text{inside}\text{\hspace{0.17em}}\text{them}\text{.}\hfill \\ 5{y}^{3}+11-12{y}^{3}+2\hfill & \text{Combine}\text{\hspace{0.17em}}\text{like}\text{\hspace{0.17em}}\text{terms}\text{.}\hfill \\ -7{y}^{3}+13\hfill & \hfill \end{array}$
Add $4{x}^{2}+2x-8$ to $3{x}^{2}-7x-10$ .
$$\begin{array}{l}(4{x}^{2}+2x-8)+(3{x}^{2}-7x-10)\\ 4{x}^{2}+2x-8+3{x}^{2}-7x-10\\ 7{x}^{2}-5x-18\end{array}$$
Subtract $8{x}^{2}-5x+2$ from $3{x}^{2}+x-12$ .
$$\begin{array}{l}(3{x}^{2}+x-12)-(8{x}^{2}-5x+2)\\ 3{x}^{2}+x-12-8{x}^{2}+5x-2\\ -5{x}^{2}+6x-14\end{array}$$
Be very careful not to write this problem as
$$3{x}^{2}+x-12-8{x}^{2}-5x+2$$
This form has us subtracting only the very first term,
$8{x}^{2}$ , rather than the entire expression. Use parentheses.
Another incorrect form is
$$8{x}^{2}-5x+2-(3{x}^{2}+x-12)$$
This form has us performing the subtraction in the wrong order.
Perform the following additions and subtractions.
For the following problems, perform the multiplications and combine any like terms.
$2(x-10)$
$6(3x+4)$
$5(8m-6)$
$-3(b+8)$
$-6(y+7)$
$-9(k-7)$
$-7(4x+2)$
$-8(4y-11)$
$k(k-11)$
$4y(y+7)$
$9x(x-3)$
$4m(2m+7)$
$7a(a-4)$
$9{y}^{3}(3{y}^{2}+2)$
$4{a}^{4}(5{a}^{3}+3{a}^{2}+2a)$
$20{a}^{7}+12{a}^{6}+8{a}^{5}$
$2{x}^{4}(6{x}^{3}-5{x}^{2}-2x+3)$
$-6{y}^{3}(y+5)$
$2{x}^{2}y(3{x}^{2}{y}^{2}-6x)$
$6{x}^{4}{y}^{3}-12{x}^{3}y$
$8{a}^{3}{b}^{2}c(2a{b}^{3}+3b)$
$4x(3{x}^{2}-6x+10)$
$9{y}^{3}(2{y}^{4}-3{y}^{3}+8{y}^{2}+y-6)$
$18{y}^{7}-27{y}^{6}+72{y}^{5}+9{y}^{4}-54{y}^{3}$
$-{a}^{2}{b}^{3}(6a{b}^{4}+5a{b}^{3}-8{b}^{2}+7b-2)$
$(x+1)(x+7)$
$(t+8)(t-2)$
$(x-y)(2x+y)$
$(5a-2)(6a-8)$
$(2t+6)(3t+4)$
$(6+a)(4+a)$
$({x}^{2}+5)(x+4)$
$(4{a}^{2}{b}^{3}-2a)(5{a}^{2}b-3b)$
$(6{x}^{3}{y}^{4}+6x)(2{x}^{2}{y}^{3}+5y)$
$12{x}^{5}{y}^{7}+30{x}^{3}{y}^{5}+12{x}^{3}{y}^{3}+30xy$
$5(x-7)(x-3)$
$a(a-3)(a+5)$
${x}^{2}(x+5)(x+7)$
$2{a}^{2}(a+4)(a+3)$
$a{b}^{2}({a}^{2}-2b)(a+{b}^{4})$
${x}^{3}{y}^{2}(5{x}^{2}{y}^{2}-3)(2xy-1)$
$10{x}^{6}{y}^{5}-5{x}^{5}{y}^{4}-6{x}^{4}{y}^{3}+3{x}^{3}{y}^{2}$
$6({a}^{2}+5a+3)$
$3{a}^{2}(2{a}^{3}-10{a}^{2}-4a+9)$
$6{a}^{3}{b}^{3}(4{a}^{2}{b}^{6}+7a{b}^{8}+2{b}^{10}+14)$
$24{a}^{5}{b}^{9}+42{a}^{4}{b}^{11}+12{a}^{3}{b}^{13}+18{a}^{3}{b}^{3}$
$(a-4)({a}^{2}+a-5)$
$(2x+1)(5{x}^{3}+6{x}^{2}+8)$
$(7{a}^{2}+2)(3{a}^{5}-4{a}^{3}-a-1)$
$21{a}^{7}-22{a}^{5}-15{a}^{3}-7{a}^{2}-2a-2$
$(x+y)(2{x}^{2}+3xy+5{y}^{2})$
$(2a+b)(5{a}^{2}+4{a}^{2}b-b-4)$
$10{a}^{3}+8{a}^{3}b+4{a}^{2}{b}^{2}+5{a}^{2}b-{b}^{2}-8a-4b-2ab$
${(x+3)}^{2}$
${(x-5)}^{2}$
${(a-9)}^{2}$
$-{(8t+7)}^{2}$
For the following problems, perform the indicated operations and combine like terms.
$-2{x}^{3}+4{x}^{2}+5x-8+({x}^{3}-3{x}^{2}-11x+1)$
$(6{a}^{2}-3a+7)-4{a}^{2}+2a-8$
$(3{x}^{3}-7{x}^{2}+2)+({x}^{3}+6)$
$(9{a}^{2}b-3ab+12a{b}^{2})+a{b}^{2}+2ab$
$9{a}^{2}b+13a{b}^{2}-ab$
$6{x}^{2}-12x+(4{x}^{2}-3x-1)+4{x}^{2}-10x-4$
$5{a}^{3}-2a-26+(4{a}^{3}-11{a}^{2}+2a)-7a+8{a}^{3}+20$
$17{a}^{3}-11{a}^{2}-7a-6$
$2xy-15-(5xy+4)$
Add $5{y}^{2}-5y+1$ to $-9{y}^{2}+4y-2$ .
Add $-2({x}^{2}-4)$ to $5({x}^{2}+3x-1)$ .
Add five times $-3x+2$ to seven times $4x+3$ .
Subtract $6{x}^{2}-10x+4$ from $3{x}^{2}-2x+5$ .
( [link] ) Simplify ${\left(\frac{15{x}^{2}{y}^{6}}{5x{y}^{2}}\right)}^{4}$ .
( [link] ) Express the number 198,000 using scientific notation.
$1.98\times {10}^{5}$
( [link] ) How many $4{a}^{2}{x}^{3}\text{'}\text{s}$ are there in $-16{a}^{4}{x}^{5}$ ?
( [link] ) State the degree of the polynomial $4x{y}^{3}+3{x}^{5}y-5{x}^{3}{y}^{3}$ , and write the numerical coefficient of each term.
$\text{degree}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}6;\text{\hspace{0.17em}}\text{\hspace{0.17em}}4,3,-5$
( [link] ) Simplify $3(4x-5)+2(5x-2)-(x-3)$ .
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