// Numbas version: finer_feedback_settings {"name": "Expanding a binomial product (monic factors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Expanding a binomial product (monic factors)", "tags": ["binomial", "Binomial", "binomial product", "distributive law", "expanding", "Expanding", "factorisation", "Factorisation", "factors", "Factors", "monic", "quadratic"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Expand and simplify the following.

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

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Method 1 (the distributive law)

We expand $\\simplify[basic]{(x+{a[0]})(x+{b[0]})}$ one bracket at a time. Each term in the first bracket times the entire other bracket: $\\simplify[basic]{x(x+{b[0]})+{a[0]}(x+{b[0]})}$

Then we use the distributive law on each bracket: $\\simplify[basic, !collectnumbers]{x^2+{b[0]}x+{a[0]}x+{a[0]*b[0]}}$

And collect like terms: $\\simplify[basic, unitfactor]{x^2+{a[0]+b[0]}x+{a[0]*b[0]}}$

Method 2 (FOIL)

Multiply the First terms in each bracket to get $x^2$, then the Outer terms to get $\\var{b[0]}x$, then the Inner terms to get $\\var{a[0]}x$, and then the Last terms to get $\\var{a[0]*b[0]}$. Now add them all together: $\\simplify[basic, unitfactor]{x^2+{a[0]+b[0]}x+{a[0]*b[0]}}$

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Ensure you don't use brackets in your answer.

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$\\simplify{(x+{a[1]})(x+{b[1]})}$ = [[0]]

Method 1 (the distributive law)

We expand $\\simplify[basic]{(x+{a[1]})(x+{b[1]})}$ one bracket at a time. Each term in the first bracket times the entire other bracket: $\\simplify[basic]{x(x+{b[1]})+{a[1]}(x+{b[1]})}$

Then we use the distributive law on each bracket: $\\simplify[basic, !collectnumbers]{x^2+{b[1]}x+{a[1]}x+{a[1]*b[1]}}$

And collect like terms: $\\simplify[basic, unitfactor]{x^2+{a[1]+b[1]}x+{a[1]*b[1]}}$

Method 2 (FOIL)

Multiply the First terms in each bracket to get $x^2$, then the Outer terms to get $\\var{b[1]}x$, then the Inner terms to get $\\var{a[1]}x$, and then the Last terms to get $\\var{a[1]*b[1]}$. Now add them all together: $\\simplify[basic, unitfactor]{x^2+{a[1]+b[1]}x+{a[1]*b[1]}}$

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Ensure you don't use brackets in your answer.

$\\simplify{(m+{a[2]})(m+{b[2]})}$ = [[0]]

Method 1 (the distributive law)

We expand $\\simplify[basic]{(m+{a[2]})(m+{b[2]})}$ one bracket at a time. Each term in the first bracket times the entire other bracket: $\\simplify[basic]{m(m+{b[2]})+{a[2]}(m+{b[2]})}$

Then we use the distributive law on each bracket: $\\simplify[basic, !collectnumbers]{m^2+{b[2]}m+{a[2]}m+{a[2]*b[2]}}$

And collect like terms: $\\simplify[basic, unitfactor]{m^2+{a[2]+b[2]}m+{a[2]*b[2]}}$

Method 2 (FOIL)

Multiply the First terms in each bracket to get $m^2$, then the Outer terms to get $\\var{b[2]}m$, then the Inner terms to get $\\var{a[2]}m$, and then the Last terms to get $\\var{a[2]*b[2]}$. Now add them all together: $\\simplify[basic, unitfactor]{m^2+{a[2]+b[2]}m+{a[2]*b[2]}}$

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Ensure you don't use brackets in your answer.

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$\\simplify{(t+{a[3]})(t+{b[3]})}$ = [[0]]

Method 1 (the distributive law)

We expand $\\simplify[basic]{(t+{a[3]})(t+{b[3]})}$ one bracket at a time. Each term in the first bracket times the entire other bracket: $\\simplify[basic]{t(t+{b[3]})+{a[3]}(t+{b[3]})}$

Then we use the distributive law on each bracket: $\\simplify[basic, !collectnumbers]{t^2+{b[3]}t+{a[3]}t+{a[3]*b[3]}}$

And collect like terms: $\\simplify[basic, unitfactor]{t^2+{a[3]+b[3]}t+{a[3]*b[3]}}$

Method 2 (FOIL)

Multiply the First terms in each bracket to get $t^2$, then the Outer terms to get $\\var{b[3]}t$, then the Inner terms to get $\\var{a[3]}t$, and then the Last terms to get $\\var{a[3]*b[3]}$. Now add them all together: $\\simplify[basic, unitfactor]{t^2+{a[3]+b[3]}t+{a[3]*b[3]}}$

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Ensure you don't use brackets in your answer.

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