// Numbas version: exam_results_page_options {"name": "Expanding a binomial product (monic factors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "b"], "tags": ["binomial", "binomial product", "distributive law", "expanding", "Factorisation", "factorisation", "Factors", "factors", "monic", "quadratic"], "name": "Expanding a binomial product (monic factors)", "statement": "

Expand and simplify the following.

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

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(x+{a[0]})(x+{b[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+{b[0]})}\$ \$=\$ \n\$\\simplify{x(x+{b[0]})+{a[0]}(x+{b[0]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{x^2+{b[0]}x+{a[0]}x+{a[0]*b[0]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{x^2+{b[0]+a[0]}x+{a[0]*b[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+{b[0]})}\$ \$=\$ \n\$\\simplify[basic]{x^2+{b[0]}x+{a[0]}x+{a[0]*b[0]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{x^2+{b[0]+a[0]}x+{a[0]*b[0]}}\$ (collect like terms)
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\$\\simplify{(x+{a[1]})(x+{b[1]})}\$ = [[0]]

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(x+{a[1]})(x+{b[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{(x+{a[1]})(x+{b[1]})}\$ \$=\$ \n\$\\simplify{x(x+{b[1]})+{a[1]}(x+{b[1]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{x^2+{b[1]}x+{a[1]}x+{a[1]*b[1]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{x^2+{b[1]+a[1]}x+{a[1]*b[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{(x+{a[1]})(x+{b[1]})}\$ \$=\$ \n\$\\simplify[basic]{x^2+{b[1]}x+{a[1]}x+{a[1]*b[1]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{x^2+{b[1]+a[1]}x+{a[1]*b[1]}}\$ (collect like terms)
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\$\\simplify{(m+{a[2]})(m+{b[2]})}\$ = [[0]]

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(m+{a[2]})(m+{b[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{(m+{a[2]})(m+{b[2]})}\$ \$=\$ \n\$\\simplify{m(m+{b[2]})+{a[2]}(m+{b[2]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{m^2+{b[2]}m+{a[2]}m+{a[2]*b[2]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{m^2+{b[2]+a[2]}m+{a[2]*b[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{(m+{a[2]})(m+{b[2]})}\$ \$=\$ \n\$\\simplify[basic]{m^2+{b[2]}m+{a[2]}m+{a[2]*b[2]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{m^2+{b[2]+a[2]}m+{a[2]*b[2]}}\$ (collect like terms)
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\$\\simplify{(t+{a[3]})(t+{b[3]})}\$ = [[0]]

\n

\n

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Method 1 (the distributive law)

\n

We expand \$\\simplify{(t+{a[3]})(t+{b[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{(t+{a[3]})(t+{b[3]})}\$ \$=\$ \n\$\\simplify{t(t+{b[3]})+{a[3]}(t+{b[3]})}\$\n \n          (each term in one bracket times the entire other bracket)\n \$=\$ \$\\simplify{t^2+{b[3]}t+{a[3]}t+{a[3]*b[3]}}\$ (use the distributive law on each bracket) \$=\$ \$\\simplify{t^2+{b[3]+a[3]}t+{a[3]*b[3]}}\$ \n\n          (collect like terms)\n
\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{(t+{a[3]})(t+{b[3]})}\$ \$=\$ \n\$\\simplify[basic]{t^2+{b[3]}t+{a[3]}t+{a[3]*b[3]}}\$\n \n          (First, Outer, Inner, Last)\n \$=\$ \$\\simplify{t^2+{b[3]+a[3]}t+{a[3]*b[3]}}\$ (collect like terms)
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