// Numbas version: exam_results_page_options {"name": "Polynomial long division: degree of remainder and quotient", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["n", "c", "m"], "name": "Polynomial long division: degree of remainder and quotient", "tags": ["Factors", "factors", "polynomial long division", "polynomials", "remainders", "roots"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "displayColumns": 0, "prompt": "

Suppose you divide a polynomial of degree $\\var{n}$ by $\\simplify{x-{c}}$, then the remainder must be a

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constant.

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linear function.

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quadratic function.

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polynomial of degree $\\var{n-1}$.

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When you divide the dividend by the divisor, the remainder must be a polynomial of a smaller degree than the divisor.

In the case that the divisor is $\\simplify{x-{c}}$ (a polynomial of degree 1) the remainder must be a polynomial of 0th degree, that is, a constant.

Suppose you divide a polynomial of degree $\\var{n}$ by a polynomial of degree $\\var{m}$, then we can say the remainder is a polynomial of degree no more than

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$\\var{m}$

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$\\var{m-1}$

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$\\var{m-2}$

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$0$

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When you divide the dividend by the divisor, the remainder must be a polynomial of a smaller degree than the divisor.

In the case that the divisor is a polynomial of degree $\\var{m}$ the remainder must be a polynomial of degree $\\var{m-1}$ or less.

Again, suppose you divide a polynomial of degree $\\var{n}$ by a polynomial of degree $\\var{m}$, then we can say the quotient is a polynomial of degree

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$\\var{n-m}$

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$\\var{n-m-2}$

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$\\var{n-m-1}$

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$\\var{n-m+1}$

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$\\var{n}$

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Due to the index laws, the degree of the dividend must equal the sum of the degrees of the divisor and quotient.

This means that the degree of the quotient must be $\\var{n}-\\var{m}=\\var{n-m}$.

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