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Expand and simplify the following.

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(x+{a})(x-{a})}\$ 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})(x-{a})}\$ \$=\$ \n\$\\simplify{x(x-{a})+{a}(x-{a})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{x^2-{a}x+{a}x-{a*a}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{x^2-{a*a}}\$ (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})(x-{a})}\$ \$=\$ \n\$\\simplify[basic]{x^2-{a}x+{a}x-{a*a}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{x^2-{a*a}}\$ (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})(x-{a})}\$ \$=\$ \n\$\\simplify{x^2-{a*a}}\$\n \n          (difference of two squares)\n
\n

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\$\\simplify{(x+{a})(x-{a})}\$ = []

\n

\n

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

\n

\n

Method 1 (the distributive law)

\n

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

Method 3 (perfect square)

\n

Notice that \$\\simplify{(x+{a})^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})}\$ \$=\$ \n\$\\simplify{x^2+{2*a}x+{a*a}}\$\n \n          (perfect square)\n
", "type": "information", "unitTests": [], "variableReplacements": []}], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

\$\\simplify{(x+{a})^2}\$ = []

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(w+{a})(w-{a})}\$ 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})(w-{a})}\$ \$=\$ \n\$\\simplify{w(w-{a})+{a}(w-{a})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{w^2-{a}w+{a}w-{a*a}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{w^2-{a*a}}\$ (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})(w-{a})}\$ \$=\$ \n\$\\simplify[basic]{w^2-{a}w+{a}w-{a*a}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{w^2-{a*a}}\$ (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})(w-{a})}\$ \$=\$ \n\$\\simplify{w^2-{a*a}}\$\n \n          (difference of two squares)\n
", "type": "information", "unitTests": [], "variableReplacements": []}], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

\$\\simplify{(w+{a})(w-{a})}\$ = []

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(r+{a})(r+{a})}\$ 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})(r+{a})}\$ \$=\$ \n\$\\simplify{r(r+{a})+{a}(r+{a})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{r^2+{a}r+{a}r+{a*a}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{r^2+{2*a}r+{a*a}}\$ (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})(r+{a})}\$ \$=\$ \n\$\\simplify[basic]{r^2+{a}r+{a}r+{a*a}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{r^2+{2*a}r+{a*a}}\$ (collect like terms)
\n

Method 3 (perfect square)

\n

Notice that \$\\simplify{(r+{a})(r+{a})}\$ 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})(r+{a})}\$ \$=\$ \n\$\\simplify{r^2+{2*a}r+{a*a}}\$\n \n          (perfect square)\n
", "type": "information", "unitTests": [], "variableReplacements": []}], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

\$\\simplify{(r+{a})(r+{a})}\$ = []

\n

\n

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