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SingaporeAdditional MathematicsQuick questions

Algebra: Surds, Indices and Polynomials

Quick questions on Remainder and factor theorems: N(A)-Level Additional Mathematics

8short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the remainder theorem?
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When a polynomial f(x)f(x) is divided by (x−a)(x - a), the remainder is simply the value f(a)f(a):
What is the factor theorem?
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The factor theorem is the special case where the remainder is zero:
What is factorising a cubic?
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To factorise a cubic f(x)f(x) fully:
What is finding an unknown coefficient?
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If a polynomial contains an unknown and you are told a remainder or a factor, set up an equation. "(x−a)(x - a) is a factor" gives f(a)=0f(a) = 0; "the remainder on division by (x−a)(x - a) is rr" gives f(a)=rf(a) = r. Solve for the unknown.
What is algebra slips in long division?
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Comparing coefficients is often safer than long division and gives an easy self-check.
What is q1?
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Find the remainder when x3+x−5x^3 + x - 5 is divided by (x−1)(x - 1). [2 marks]
What is q2?
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Show that (x+2)(x + 2) is a factor of x3+3x2−4x^3 + 3x^2 - 4. [2 marks]
What is q3?
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Given that (x−3)(x - 3) is a factor of x3−2x2+kx−3x^3 - 2x^2 + kx - 3, find kk. [3 marks]

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