Introduction to Polynomials Class 9: notes and important questions

Introduction to Polynomials: revision notes

1. Expressions and polynomials

An algebraic expression combines numbers and variables with \(+, -, \times, \div\). It is a polynomial in \(x\) when every power of \(x\) is a whole number \((0, 1, 2, \ldots)\): \(4x^3 - x + 7\) is a polynomial, \(x + \dfrac1x\) and \(\sqrt x + 1\) are not.

• The degree is the highest power of the variable with a non-zero coefficient.
• Degree 1: linear (\(ax + b,\ a \neq 0\)); degree 2: quadratic; degree 3: cubic. A non-zero constant has degree 0.

2. Value and zero of a linear polynomial

The value of \(p(x) = ax + b\) at \(x = c\) is \(p(c) = ac + b\). The zero of \(p\) is the value of \(x\) that makes \(p(x) = 0\): \(x = -\dfrac ba\). A linear polynomial has exactly one zero.

3. Linear growth and decay

A quantity that changes by the same amount in each equal step (per hour, per km, per month) is modelled by \(y = ax + b\): \(b\) is the starting value (at \(x = 0\)) and \(a\) the change per step.

• \(a \gt 0\): linear growth (a taxi fare, savings). \(a \lt 0\): linear decay (a burning candle, a draining tank).
• A table of values is linear exactly when the differences between consecutive \(y\)-values are equal for equal steps in \(x\).

4. Graph of \(y = ax + b\): slope and intercept

The graph is a straight line. The slope is \(a\): the rise in \(y\) for each unit increase in \(x\). Through two points, slope \(= \dfrac{y_2 - y_1}{x_2 - x_1}\). The y-intercept is \(b\): the line crosses the \(y\)-axis at \((0, b)\); it crosses the \(x\)-axis at the zero, \(\left(-\dfrac ba, 0\right)\).

Worked example 1

Find the linear polynomial \(p(x)\) with \(p(0) = 4\) and \(p(3) = 13\).

\(b = 4\) and \(3a + 4 = 13 \Rightarrow a = 3\), so \(p(x) = 3x + 4\).

Worked example 2

A phone battery at \(100\%\) loses \(8\%\) every hour of video. When is it at \(20\%\)?

\(B = 100 - 8t\); \(100 - 8t = 20 \Rightarrow t = 10\) hours.

Worked example 3

Find the slope and the intercepts of \(y = 6 - 2x\).

Slope \(-2\) (decay), \(y\)-intercept \((0, 6)\), zero \(x = 3\), so it meets the \(x\)-axis at \((3, 0)\).

Common errors

• Calling \(x + \dfrac1x\) or \(\sqrt x\) a polynomial: the powers must be whole numbers.
• Reading the slope of \(y = 5 - 3x\) as \(5\): the slope is the coefficient of \(x\), here \(-3\).
• Slope as \(\dfrac{x_2 - x_1}{y_2 - y_1}\) (upside down), or mixing the order of the points in numerator and denominator.
• In word problems, using the value after one step as the starting value \(b\).

Exam tips

• In a model, say what \(x\) and \(y\) stand for, with units, before writing the formula.
• Check a model by substituting one value from the question.