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NSWMaths Extension 1Quick questions

Functions (ME-F1, ME-F2)

Quick questions on Polynomial division and the remainder and factor theorems

11short 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 division algorithm for polynomials?
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If P(x)P(x) is a polynomial of degree nn and D(x)D(x) is a polynomial of degree k≤nk \le n, then there exist unique polynomials Q(x)Q(x) (the quotient) of degree n−kn - k and R(x)R(x) (the remainder) of degree less than kk such that
What is the remainder theorem?
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If P(x)P(x) is divided by (x−a)(x - a), the remainder is P(a)P(a). In other words,
What is the factor theorem?
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The factor theorem is the special case of the remainder theorem where the remainder is zero.
What is long division of polynomials?
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To divide P(x)P(x) by D(x)D(x), set up the long-division layout and at each step divide the leading term of the current dividend by the leading term of the divisor, multiply back, and subtract.
What is strategy for factorising a cubic?
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To fully factorise a cubic P(x)P(x) over the reals:
What is full factorisation?
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Factorise P(x)=x3−6x2+11x−6P(x) = x^3 - 6 x^2 + 11 x - 6.
What is polynomial long division?
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Divide P(x)=2x3+x−4P(x) = 2 x^3 + x - 4 by D(x)=x2−1D(x) = x^2 - 1.
What is forgetting placeholder zeros?
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When long-dividing, you must include zero coefficients for missing powers. x3+1x^3 + 1 is really x3+0x2+0x+1x^3 + 0 x^2 + 0 x + 1.
What is using the wrong root from a factor?
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(2x−1)(2 x - 1) is a factor iff P(1/2)=0P(1/2) = 0, not P(1)P(1) or P(2)P(2).
What is not testing the rational root candidates systematically?
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For P(x)=2x3+3x2−8x+3P(x) = 2 x^3 + 3 x^2 - 8 x + 3, candidates are ±1,±3,±12,±32\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}. Try in order and confirm with the factor theorem.
What is stopping at the linear divisor?
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A cubic factorises as a product of three linear factors over R\mathbb{R} only if all roots are real. If the quadratic quotient has no real roots, leave it as a quadratic factor. :::

All Maths Extension 1Q&A pages