Linear, quadratic and polynomial relationships
| English | Español |
|---|---|
| sign interval/saɪn ˈɪntəvl/ | sign interval |
| polynomial degree/ˌpɒlɪˈnəʊmɪəl dɪˈɡriː/ | polynomial degree |
A decision before an answer
- The roots of a quadratic mark boundaries. They do not by themselves tell you which interval makes the expression negative.
- Your goal: Choose equivalent expanded, factored or vertex forms for a task.
Read the relationship
- Translate relationships into expressions before manipulating them. A fixed fee plus a per-unit charge is an affine linear model, while a product of dimensions can create a quadratic. An equivalent form should preserve the original expression on its domain. Factor common terms and recognise identities before expanding every product.
- Solve linear and polynomial equations using structure.
x(x-5)=0 has solutions:
Either factor may be zero.
Use the defining rule
- A linear equation can have one, no or infinitely many solutions after simplifying; check zero coefficients before dividing. Multiplying or dividing an inequality by a negative reverses its direction. A polynomial product equals zero when at least one factor is zero. Keep all factors, including an extracted x, to avoid losing a zero root.
- Determine quadratic inequality regions from signs rather than roots alone.
(x-1)(x-4)<0 holds on:
The factors have opposite signs strictly between the roots.
Check the conditions
- A quadratic’s factored form exposes roots and its vertex form exposes an extremum. For an inequality, order the real roots and test the sign in each interval. Include boundaries only for ≤ or ≥. A parabola opening upward is negative between two distinct real roots and positive outside them.
- Determine quadratic inequality regions from signs rather than roots alone.
x²-5x+6=(x-2)(x-3), so x²-5x+6<0 gives 2<x<3. x³-4x=x(x-2)(x+2), giving roots -2,0,2. The expression 2(x-3)²+5 has minimum 5 at x=3. In (2x+1)(x-4), the x coefficient is -8+1=-7.
The minimum of (x+2)²+7 is ____.
The squared term is nonnegative and can equal zero.
Apply the task format
- Use substitution into the original relation to check candidate values. An expanded coefficient question may require combining several contributions to the same power. A degree-three polynomial need not require a general cubic formula: a common factor or known root can reduce it to simpler factors. Choose the method based on structure.
- Determine quadratic inequality regions from signs rather than roots alone.
Do not omit an extracted factor, include a strict-inequality boundary, or report a vertex x-coordinate when the question asks for the minimum value.
Which answer fits this case?
Choose equivalent expanded, factored or vertex forms for a task
Dividing an inequality by -2 leaves its direction unchanged.
A negative multiplier reverses order.
Keep the distinctions
- sign interval 符号区间 — A region between roots on which an expression keeps one sign.
- polynomial degree 多项式次数 — The highest variable power with a nonzero coefficient after simplification.
- Choose equivalent expanded, factored or vertex forms for a task.
- Solve linear and polynomial equations using structure.
- Determine quadratic inequality regions from signs rather than roots alone.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.