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Pure mathematics: foundations

Quick questions on Polynomials and the binomial theorem: factor theorem, division, expansion - OCR A-Level Maths A

7short 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 polynomial manipulation?
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A polynomial is a sum of terms anxn+⋯+a1x+a0a_n x^n + \dots + a_1 x + a_0. You add and subtract by collecting like terms, and multiply by expanding every pair of terms and collecting.
What is the factor theorem?
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The factor theorem links roots and factors:
What is algebraic division?
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Divide a polynomial by a linear factor using long division or by comparing coefficients. Comparing coefficients is often quicker: write f(x)=(x−a)(quadratic)f(x) = (x - a)(\text{quadratic}) with unknown coefficients and match.
What is solving an equation with a polynomial factorised?
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Once a cubic is fully factorised, its roots are immediate, which is why factorising is the standard route to solving cubic equations. The roots are exactly the values that make each factor zero.
What is sign errors with a negative term?
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In (a−b)n(a - b)^n the signs alternate; track (−1)r(-1)^r carefully.
What is q1?
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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 q2?
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Find the coefficient of x2x^2 in (1+2x)6(1 + 2x)^6. [2 marks]

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