DPST1012 Chap.5 Linear and Quadratic Polynomials
Linear and Quadratic Polynomials
What counts as a polynomial
Week 3 begins the second topic, Polynomial Functions. A polynomial adds together whole-number powers of x, each with a real coefficient, so square roots of x and negative powers are excluded. The worksheet names its parts: terms, the leading term with the highest power, the leading coefficient, the constant term and the degree.
Every polynomial has domain ℝ; one of odd degree also has range ℝ, while one of even degree has a maximum or minimum value. When two polynomials are multiplied, their degrees add.
Linear polynomials as models
A linear polynomial ax + b changes at a constant rate a, its gradient, and starts from the constant term b, its y-intercept.
The worksheet interprets these in context, from a phone plan with a base fee and a cost per call to a leaking tank losing a fixed volume each hour.
In context, the domain shrinks to the values that make sense, such as times from zero until the tank is empty.
Quadratics: solving and sketching
The course motivates quadratics with a parabola fitted to Sydney's mean maximum temperature from April to September, which leads to equations of the form ax² + bx + c = 0. Three methods are taught: factorisation when the roots are simple integers, completing the square, and the quadratic formula, whose discriminant b² − 4ac decides whether there are two, one or no real roots.
Sketching combines the sign of a, the roots, the y-intercept and the vertex at x = −b/(2a), and the sketch then solves quadratic inequalities by reading where the parabola lies above or below the axis.
What this chapter covers
- 01
Definition and features of a polynomial
- 02
Degree of a product
- 03
Domain and range by degree
- 04
Linear models: rate of change and constant term
- 05
Factorising quadratics
- 06
Completing the square
- 07
The discriminant and the quadratic formula
- 08
Vertex form and sketching
- 09
Quadratic inequalities
Worked example · free
Solve, sketch and use a quadratic in one question
- 2f(x) = −(x² − 4x − 5) = −(x − 5)(x + 1), so the roots are x = 5 and x = −1.
- 2h = −b/(2a) = −4/(−2) = 2 and f(2) = −4 + 8 + 5 = 9, so the vertex is (2, 9), a maximum because a < 0.
- 1The y-intercept is f(0) = 5.
- 1The parabola opens downwards, so it is above the axis between its roots: −1 < x < 5.
Key terms
- Leading Coefficient
- The number multiplying the highest power of x in a polynomial, which must not be zero.
- Degree
- The highest power of x that appears in a polynomial with a non-zero coefficient.
- Discriminant
- The quantity b² − 4ac, whose sign tells how many real roots a quadratic equation has.
- Vertex Form
- The way of writing a quadratic as a(x − h)² + k, so that its turning point (h, k) can be read directly.
- Gradient
- The rate of change of a linear function, equal to the change in output for each one-unit increase in input.
Linear and Quadratic Polynomials FAQ
Which method should I use to solve a quadratic?
Try factorising when the leading coefficient is 1 and the numbers are small. Complete the square when you also want the vertex. Use the formula whenever the numbers are awkward, because it always works.
What does a negative discriminant tell me?
That the quadratic equation has no real solutions, so its parabola never meets the x-axis. If the parabola opens upwards it lies entirely above the axis; if downwards, entirely below.
Why is the domain of a polynomial all real numbers?
A polynomial only adds, subtracts and multiplies, and those operations work for every real number. Nothing can go wrong at any input, unlike division by zero or a negative square root.
How do I interpret the gradient of a linear model in words?
Name the output, the unit and the direction for a one-unit increase in the input, for example 'the volume falls by 25 litres every hour'. A bare number without units rarely earns the interpretation mark.
How do I find a quadratic from its graph?
If the roots are visible, start from a(x − α)(x − β) and use the y-intercept to find a. If the vertex is visible, start from a(x − h)² + k and substitute one more point. Either way, expand at the end to check the constant term.
Exam move
Solve the same quadratic three ways in one sitting until each method gives the same roots, then sketch it and solve both of its inequalities from the sketch. Time yourself, because the Week 4 test leaves little room for slow algebra.
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