Important Questions of Polynomials (Part 1)

This lesson runs through the key short-answer questions on polynomials: the meaning of degree, the forms and graphs of linear, quadratic, and cubic polynomials, the link between the zeros of a quadratic and its coefficients, and the standard identities you reuse again and again.

Degree of a polynomial

The exponent of the variable in a polynomial is always a whole number. The degree of a polynomial is the highest power of the variable among the terms with a non-zero coefficient.

A constant such as $P(x) = 7$ has degree $0$, because it can be seen as $7x^{0}$. The degree of the zero polynomial is not defined.

Linear polynomials

The general form of a linear polynomial is

$P(x) = ax + b, \quad a \neq 0,$

where $a$ and $b$ are real numbers. Its degree is $1$ and its graph is a straight line.

A zero of a polynomial is a value of $x$ for which $P(x) = 0$. For the linear polynomial $ax + b$, setting $P(x) = 0$ gives

$x = -\frac{b}{a}.$

So a linear polynomial has exactly one zero.

Quadratic polynomials

The general form of a quadratic polynomial is

$P(x) = ax^{2} + bx + c, \quad a \neq 0,$

with $a, b, c$ real. Its graph is a parabola.

Direction of the parabola

• If $a > 0$, the parabola opens upward.
• If $a < 0$, the parabola opens downward.

A quadratic polynomial has at most $2$ zeros.

Zeros and coefficients

If $\alpha$ and $\beta$ are the zeros of $ax^{2} + bx + c$, then

$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a},$

where $a$ is the coefficient of $x^{2}$, $b$ the coefficient of $x$, and $c$ the constant term. In words, the sum of the zeros is minus the coefficient of $x$ over the coefficient of $x^{2}$, and the product of the zeros is the constant over the coefficient of $x^{2}$.

Rebuilding the quadratic from its zeros

Given the zeros $\alpha$ and $\beta$, the quadratic can be written as

$x^{2} - (\alpha + \beta)x + \alpha\beta,$

that is, $x^{2} - (\text{sum of zeros})x + (\text{product of zeros})$.

Cubic polynomials

A cubic polynomial has the general form $ax^{3} + bx^{2} + cx + d$ with $a \neq 0$, and it has $3$ zeros.

Checking whether a value is a zero

To check whether a given value is a zero of a polynomial, substitute it and see if the result is $0$.

Take $P(x) = x^{2} - 4$ and check whether $2$ is a zero. Put $x = 2$:

$P(2) = 2^{2} - 4 = 4 - 4 = 0.$

Since $P(2) = 0$, the value $2$ is a zero of $P(x)$.

Useful identities

If $\alpha$ and $\beta$ are the zeros of a quadratic $ax^{2} + bx + c$, then alongside

$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a},$

the following identities are often needed:

$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta},$

$\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta,$

$\alpha^{4} + \beta^{4} = \left(\alpha^{2} + \beta^{2}\right)^{2} - 2(\alpha\beta)^{2},$

$\alpha^{3} + \beta^{3} = (\alpha + \beta)^{3} - 3\alpha\beta(\alpha + \beta).$

Key takeaways

• The degree is the highest power with a non-zero coefficient; a constant has degree $0$ and the zero polynomial has undefined degree.
• Linear, quadratic, and cubic polynomials have $1$, $2$, and $3$ zeros, with graphs that are a line and a parabola respectively.
• For a quadratic, $\alpha + \beta = -\tfrac{b}{a}$ and $\alpha\beta = \tfrac{c}{a}$, and the symmetric identities above let you evaluate expressions in $\alpha$ and $\beta$ without solving for them.