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Expand and simplify the following.

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(x+{a[0]})(x-{a[0]})}\$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[0]})(x-{a[0]})}\$ \$=\$ \n\$\\simplify{x(x-{a[0]})+{a[0]}(x-{a[0]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{x^2-{a[0]}x+{a[0]}x-{a[0]*a[0]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{x^2-{a[0]*a[0]}}\$ (collect like terms)
\n

Method 2 (FOIL)

\n

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[0]})(x-{a[0]})}\$ \$=\$ \n\$\\simplify[basic]{x^2-{a[0]}x+{a[0]}x-{a[0]*a[0]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{x^2-{a[0]*a[0]}}\$ (collect like terms)
\n

Method 3 (difference of two squares)

\n

Notice that the product will expand to be a difference of two squares. Square the first term minus the square of the second term.

\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[0]})(x-{a[0]})}\$ \$=\$ \n\$\\simplify{x^2-{a[0]*a[0]}}\$\n \n          (difference of two squares)\n
\n

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\$\\simplify{(x+{a[0]})(x-{a[0]})}\$ = [[0]]

\n

\n

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It is important to realise that \$\\simplify{(x+{a[2]})^2}=\\simplify{(x+{a[2]})(x+{a[2]})}\$. Recall that squaring something is multiplying it by itself.

\n

\n

Method 1 (the distributive law)

\n

We expand \$\\simplify{(x+{a[2]})(x+{a[2]})}\$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[2]})(x+{a[2]})}\$ \$=\$ \n\$\\simplify{x(x+{a[2]})+{a[2]}(x+{a[2]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{x^2+{a[2]}x+{a[2]}x+{a[2]*a[2]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}\$ (collect like terms)
\n

Method 2 (FOIL)

\n

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[2]})(x+{a[2]})}\$ \$=\$ \n\$\\simplify[basic]{x^2+{a[2]}x+{a[2]}x+{a[2]*a[2]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}\$ (collect like terms)
\n

Method 3 (perfect square)

\n

Notice that \$\\simplify{(x+{a[2]})^2}\$ is a perfect square. Square the first term, double the second term times the first, then square the last term, add them all together.

\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(x+{a[2]})}\$ \$=\$ \n\$\\simplify{x^2+{2*a[2]}x+{a[2]*a[2]}}\$\n \n          (perfect square)\n
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\$\\simplify{(x+{a[2]})^2}\$ = [[0]]

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(w+{a[1]})(w-{a[1]})}\$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(w+{a[1]})(w-{a[1]})}\$ \$=\$ \n\$\\simplify{w(w-{a[1]})+{a[1]}(w-{a[1]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{w^2-{a[1]}w+{a[1]}w-{a[1]*a[1]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{w^2-{a[1]*a[1]}}\$ (collect like terms)
\n

Method 2 (FOIL)

\n

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(w+{a[1]})(w-{a[1]})}\$ \$=\$ \n\$\\simplify[basic]{w^2-{a[1]}w+{a[1]}w-{a[1]*a[1]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{w^2-{a[1]*a[1]}}\$ (collect like terms)
\n

Method 3 (difference of two squares)

\n

Notice that the product will expand to be a difference of two squares. Square the first term minus the square of the second term.

\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(w+{a[1]})(w-{a[1]})}\$ \$=\$ \n\$\\simplify{w^2-{a[1]*a[1]}}\$\n \n          (difference of two squares)\n
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\$\\simplify{(w+{a[1]})(w-{a[1]})}\$ = [[0]]

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(r+{a[3]})(r+{a[3]})}\$ one bracket at a time.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(r+{a[3]})(r+{a[3]})}\$ \$=\$ \n\$\\simplify{r(r+{a[3]})+{a[3]}(r+{a[3]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{r^2+{a[3]}r+{a[3]}r+{a[3]*a[3]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}\$ (collect like terms)
\n

Method 2 (FOIL)

\n

Multiply the First terms in each bracket, then the Outer terms, then the Inner terms and then the Last terms. Add them all together.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(r+{a[3]})(r+{a[3]})}\$ \$=\$ \n\$\\simplify[basic]{r^2+{a[3]}r+{a[3]}r+{a[3]*a[3]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}\$ (collect like terms)
\n

Method 3 (perfect square)

\n

Notice that \$\\simplify{(r+{a[3]})(r+{a[3]})}\$ is a perfect square. Square the first term, double the second term times the first, then square the last term, add them all together.

\n\n\n\n\n\n\n\n\n\n
 \$\\simplify{(r+{a[3]})(r+{a[3]})}\$ \$=\$ \n\$\\simplify{r^2+{2*a[3]}r+{a[3]*a[3]}}\$\n \n          (perfect square)\n
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\$\\simplify{(r+{a[3]})(r+{a[3]})}\$ = [[0]]

\n

\n

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