Polynomials

The One Thing

The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. For quadratic and cubic polynomials, the sums and products of the zeroes can be obtained directly from the coefficients by comparing expanded and factored forms.

Definitions and Results

• Polynomial: An algebraic expression made using variables, constants, and non-negative integer powers of variables, such as $3x^2-5x+2$.

• Degree of a polynomial: The highest power of the variable with a non-zero coefficient. For example, $4x^3+x-7$ has degree $3$.

• Zero of a polynomial: A number $a$ for which $p(a)=0$. A zero is also called a root of the polynomial.

• Linear polynomial: A polynomial of degree $1$, generally written as $ax+b$, where $a\ne 0$.

• Zero of a linear polynomial: For $p(x)=ax+b$, the zero is

$$-\frac{b}{a},$$ where $a\ne 0$.

• Quadratic polynomial: A polynomial of degree $2$, generally written as

$$a{x}^2+bx+c,$$ where $a\ne 0$.

• Quadratic relationships: If $\alpha$ and $\beta$ are the zeroes of

$$p(x)=a{x}^2+bx+c,$$ then $$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$

• Quadratic polynomial constructed from its zeroes: The quadratic polynomial with zeroes $\alpha$ and $\beta$ is

$$k[x^2-(\alpha +\beta )x+\alpha \beta ],$$ where $k$ is any non-zero constant.

• Cubic polynomial: A polynomial of degree $3$, generally written as

$$a{x}^3+b{x}^2+cx+d,$$ where $a\ne 0$.

• Cubic relationships: If $\alpha ,\beta ,\gamma$ are the zeroes of

$$p(x)=a{x}^3+b{x}^2+cx+d,$$ then $$\alpha +\beta +\gamma =-\frac{b}{a},$$ $$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a},$$ and $$\alpha \beta \gamma =-\frac{d}{a}.$$

• Cubic polynomial constructed from its zeroes: The cubic polynomial with zeroes $\alpha ,\beta ,\gamma$ is

$$k[x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma ],$$ where $k$ is any non-zero constant.

• Graphical meaning of a zero: A real zero is the $x$-coordinate of a point where the graph of the polynomial intersects or touches the $x$-axis. Non-real zeroes do not appear as real $x$-axis intersections.

• Maximum number of zeroes: A polynomial of degree $n$ can have at most $n$ zeroes.

• Repeated zero: A zero may occur more than once. For example, in

$$(x-2{)}^2,$$ the zero $2$ occurs twice.

• Factor relationship: If $a$ is a zero of $p(x)$, then $(x-a)$ is a factor of $p(x)$.

Worked Methods

1. Finding the zero of a linear polynomial

For a linear polynomial $$p(x)=ax+b,$$ set the polynomial equal to zero: $$ax+b=0.$$

Rearrange: $$ax=-b.$$

Since $a\ne 0$, divide by $a$: $$x=-\frac{b}{a}.$$

Thus, the zero of $ax+b$ is $-b/a$.

2. Finding the sum and product of the zeroes of a quadratic polynomial

Let $$p(x)=a{x}^2+bx+c$$ have zeroes $\alpha$ and $\beta$.

Write the polynomial in factored form: $$p(x)=a(x-\alpha )(x-\beta ).$$

Expand: $$p(x)=a[x^2-(\alpha +\beta )x+\alpha \beta ].$$

Therefore, $$p(x)=a{x}^2-a(\alpha +\beta )x+a\alpha \beta .$$

Compare this with $$p(x)=a{x}^2+bx+c.$$

Comparing coefficients gives $$-a(\alpha +\beta )=b$$ and $$a\alpha \beta =c.$$

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

These relationships can be used without solving the quadratic completely.

3. Constructing a quadratic polynomial from its zeroes

Suppose the zeroes are $\alpha$ and $\beta$.

Begin with the factored form: $$(x-\alpha )(x-\beta ).$$

Expand: $$(x-\alpha )(x-\beta )=x^2-(\alpha +\beta )x+\alpha \beta .$$

Multiplying by any non-zero constant $k$ gives $$k[x^2-(\alpha +\beta )x+\alpha \beta ].$$

This is the general quadratic polynomial with zeroes $\alpha$ and $\beta$.

4. Finding the coefficient relationships for a cubic polynomial

Let $$p(x)=a{x}^3+b{x}^2+cx+d$$ have zeroes $\alpha ,\beta ,\gamma$.

Write the polynomial in factored form: $$p(x)=a(x-\alpha )(x-\beta )(x-\gamma ).$$

First expand the three factors: $$(x-\alpha )(x-\beta )(x-\gamma )=x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma .$$

Therefore, $$p(x)=a\left[x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma \right].$$

Comparing coefficients with $$a{x}^3+b{x}^2+cx+d$$ gives $$\alpha +\beta +\gamma =-\frac{b}{a},$$ $$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a},$$ and $$\alpha \beta \gamma =-\frac{d}{a}.$$

5. Constructing a cubic polynomial from its zeroes

Suppose the zeroes are $\alpha ,\beta ,\gamma$.

Start with the factored form: $$(x-\alpha )(x-\beta )(x-\gamma ).$$

Expanding gives $$x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma .$$

Thus, the general cubic polynomial with these zeroes is $$k[x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma ],$$ where $k\ne 0$.

6. Verifying a claimed zero

To verify that a number $a$ is a zero of $p(x)$:

• Substitute $x=a$ into the polynomial.
• Calculate $p(a)$.
• Confirm that

$$p(a)=0.$$

If the result is zero, then $a$ is a zero. In factor form, this also means that $(x-a)$ is a factor of $p(x)$.

7. Interpreting zeroes from a graph

To identify real zeroes graphically:

• Locate the points where the graph intersects or touches the $x$-axis.
• Read the $x$-coordinates of those points.
• These $x$-coordinates are the real zeroes of the polynomial.

A graph may touch the axis without crossing it, as occurs for a repeated zero such as the zero $2$ in $$(x-2{)}^2.$$

Where It Goes Wrong

• Forgetting the leading coefficient $a$ when using the relationships for $a{x}^2+bx+c$ or $a{x}^3+b{x}^2+cx+d$; the expressions must be divided by $a$.

• Losing the negative signs in

$$\alpha +\beta =-\frac{b}{a}$$ and $$\alpha +\beta +\gamma =-\frac{b}{a}.$$

• Using the incorrect sign for the cubic product: the correct relationship is

$$\alpha \beta \gamma =-\frac{d}{a}.$$

• Confusing the pairwise product

$$\alpha \beta +\beta \gamma +\gamma \alpha$$ with the total product $\alpha \beta \gamma$.

• Assuming that every zero must correspond to a crossing of the $x$-axis; a graph may touch the axis at a repeated zero, as in $(x-2{)}^2$.

• Failing to check a claimed zero by substitution. The required condition is $p(a)=0$, and the corresponding factor is $(x-a)$.

What Gets Asked

This material supports questions requiring students to:

• Find the zero of a linear polynomial $ax+b$.
• Determine the sum and product of the zeroes of a quadratic polynomial $a{x}^2+bx+c$.
• Construct a quadratic polynomial from specified zeroes.
• Determine the sum, pairwise products, and total product of the zeroes of a cubic polynomial $a{x}^3+b{x}^2+cx+d$.
• Construct a cubic polynomial from specified zeroes.
• Find an unknown zero or coefficient using the relationships between zeroes and coefficients.
• Verify a claimed zero by substitution.
• Identify real zeroes from a graph.
• Recognise repeated zeroes and interpret the factor $(x-a)$.
• Use the fact that a polynomial of degree $n$ has at most $n$ zeroes.