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NSWMaths Extension 1Quick questions

Functions (ME-F1, ME-F2)

Quick questions on Polynomial and rational inequalities: sign analysis, critical points and excluded values

14short 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 sign-table method?
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1. Factor the polynomial or rational expression completely. 2. Identify critical values: where each factor equals zero (sign-change candidates) and, for rational expressions, where the denominator is zero (excluded values).
What is critical values?
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A polynomial P(x)P(x) changes sign across each root of odd multiplicity, and does not change sign across roots of even multiplicity.
What is handling multiplicity?
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A factor like (x−2)2(x - 2)^2 touches the x-axis but does not cross. The sign is the same either side. Treat it as not changing the sign of the product.
What is rational inequalities?
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A common rookie error is to multiply both sides of AB≥0\frac{A}{B} \ge 0 by BB to clear the denominator. This is invalid because BB might be negative. Instead, move everything to one side and analyse the sign of the rational expression as a single object.
What is quadratic inequalities?
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For ax2+bx+c≥0a x^2 + b x + c \ge 0 with a>0a > 0, the parabola opens upward.
What is endpoint conventions?
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:::worked Worked example ### Quadratic inequality
What is cubic inequality?
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Factor: x(x−2)(x+2)≥0x (x - 2)(x + 2) \ge 0.
What is multiplicity changes sign behaviour?
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Solve (x−1)2(x+3)>0(x - 1)^2 (x + 3) > 0.
What is rational inequality?
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Solve x+1x−2≤0\frac{x + 1}{x - 2} \le 0.
What is moving everything to one side?
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Subtract: xx−1−1>0\frac{x}{x - 1} - 1 > 0, so x−(x−1)x−1>0\frac{x - (x - 1)}{x - 1} > 0, so 1x−1>0\frac{1}{x - 1} > 0.
What is including a zero of the denominator?
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The expression is undefined there; never include it, even in a weak inequality.
What is forgetting multiplicity?
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(x−1)2(x+2)>0(x - 1)^2 (x + 2) > 0 does not change sign at x=1x = 1. Sketching the graph helps.
What is sign sloppiness in factored form?
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If the leading coefficient is negative, every interval sign flips. Factor out the negative explicitly: −x2+3x−2=−(x−1)(x−2)-x^2 + 3 x - 2 = -(x - 1)(x - 2).
What is open versus closed endpoints in the final answer?
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Translate the inequality direction and excluded-value rules into interval notation carefully: [−2,1)∪(3,∞)[-2, 1) \cup (3, \infty) has very different meaning from (−2,1]∪[3,∞)(-2, 1] \cup [3, \infty). :::

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