// Numbas version: finer_feedback_settings {"name": "Expanding a binomial product (non-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 (non-monic factors)", "tags": ["binomial", "Binomial", "binomial product", "distributive law", "expanding", "Expanding", "factorisation", "Factorisation", "factors", "Factors", "non-monic", "quadratic"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Expand and simplify the following.

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

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Method 1 (the distributive law)

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

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

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

Method 2 (FOIL)

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

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Ensure you don't use brackets in your answer.

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

Method 1 (the distributive law)

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

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

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

Method 2 (FOIL)

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

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Ensure you don't use brackets in your answer.

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