Compare polynomial representations

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

How equations, tables, and graphs reveal different features of the same polynomial

A polynomial can be written in more than one way. The expressions

x^2-5x+6

and

(x-2)(x-3)

describe the same polynomial, but they make different features easy to notice. A table gives values, and a graph shows the shape and intercepts visually. In this lesson, you will compare these representations by asking what each one shows clearly and how the representations are connected.

What you will learn

Recognize standard, factored, and other polynomial forms.
Identify what information a representation makes easiest to see.
Use equivalent forms and selected values to compare polynomials.
Explain why a table or graph may show features less exactly than an equation.

1. Prerequisite bridge: expressions, zeros, and values

A polynomial is an expression built from numbers and whole-number powers of a variable, combined using addition, subtraction, and multiplication. For example,

2x^3-x+4

is a polynomial. A variable value can be substituted into a polynomial to calculate its output.

A zero of a polynomial is an input that makes its output equal to zero. For example, substituting

x=2

into

(x-2)(x-3)

gives zero, because one factor is zero. Zeros correspond to places where the graph meets the horizontal axis.

Two expressions are equivalent if they have the same value for every allowed input. Expanding or factoring can change how an expression looks without changing the polynomial it represents.

(x-2)(x-3)=x^2-5x+6

Substitution connects an equation to a numerical value.
A zero is an input that makes the polynomial output zero.
Equivalent expressions may look different but represent the same polynomial.

2. What different representations make visible

The standard form of a polynomial lists terms in descending powers. For a quadratic, it looks like

ax^2+bx+c

, where

a

,

b

, and

c

are numbers and

a

is not zero. Standard form makes the degree and the constant term easy to identify. The degree is the highest exponent with a nonzero coefficient.

Factored form writes a polynomial as a product of factors. For a quadratic, a form such as

a(x-r)(x-s)

makes its zeros easy to read: the zeros are

r

and

s

. Factored form can also show whether a zero is repeated, as in

a(x-r)^2

.

A table lists selected inputs and their outputs. It shows exact values at those inputs, but it does not list every possible value. A graph gives a visual picture of the outputs as the input changes. It can help show intercepts and overall shape, but a hand-drawn graph may only estimate coordinates.

To compare representations, name both the feature and the representation that shows it most directly. Do not claim that a table or sketch gives information it does not contain.

f(x)=ax^2+bx+c=a(x-r)(x-s)

Standard form makes degree and constant term easy to identify.
Factored form makes zeros easy to identify.
Tables show selected exact input-output pairs; graphs show visual behaviour.

3. Connecting equations, tables, and graphs

A representation becomes more useful when it is linked to another. Expanding a factored expression gives standard form. Substituting an input gives a table value. Plotting table pairs gives points on a graph.

For example, the factored form

(x-2)(x-3)

shows zeros at

2

and

3

. Its equivalent standard form is

x^2-5x+6

. Substituting

x=0

gives an output of

6

, so the graph crosses the vertical axis at

(0,6)

. This also matches the constant term in standard form.

A useful comparison is not just a list of forms. It explains what each form reveals and checks that the forms agree. If an equation, table, and graph appear to disagree, recalculate a value or check the plotted point.

f(0)=c

Expanding links factored form to standard form.
Substitution links an equation to a table.
Plotting input-output pairs links a table to a graph.

4. Choosing a representation for a question

Start by deciding what information the question asks for. If it asks for the degree or the constant term, standard form is convenient. If it asks for zeros, factored form is convenient. If it asks for an output at a particular input, substitution or a table is direct.

If the question asks how the polynomial looks across a range of inputs, a graph is useful. A graph can suggest where zeros occur, while a factored equation can state exact zeros when its factors are known. The strongest comparison may use both: the equation supplies exact information and the graph makes the overall behaviour visible.

A polynomial may be given in one form and need to be rewritten to answer a question. Rewriting is useful when it preserves the same polynomial and makes the requested feature clearer.

Choose the representation that makes the requested feature easiest to see.
Use another representation to confirm or add context.
A graph can suggest a feature; an equation can often state it exactly.

What each representation shows

Representation	Feature easiest to identify	Example for the same polynomial
Factored form	Zeros	$(x+1)(x-4)$ has zeros $-1$ and $4$
Standard form	Degree and constant term	$x^2-3x-4$ has degree $2$ and constant term $-4$
Table	Outputs at selected inputs	$f(0)=-4$
Graph	Visual intercepts and shape	Passes through $(-1,0)$ , $(0,-4)$ , and $(4,0)$

Worked example

Compare a quadratic in three representations

Consider

f(x)=(x+1)(x-4)

. Compare its factored form, standard form, selected table values, and graph features.

Read the factors
The factored form shows the zeros directly. Each factor is zero at its corresponding input.
$x=-1,\quad x=4$
Expand
Multiply the two binomials. This produces standard form, which makes the degree and constant term clear.
$f(x)=x^2-3x-4$
Calculate selected outputs
Substitute each listed input into the polynomial. The zeros give output zero, and the constant term gives the output at zero.
$f(-1)=0,\quad f(0)=-4,\quad f(4)=0$
Connect to the graph
The graph passes through the points from the table. It meets the horizontal axis at the zeros and the vertical axis at the output when the input is zero.
$(-1,0),\quad (0,-4),\quad (4,0)$

Answer: Factored form makes the zeros

-1

and

4

easiest to read. Standard form is

x^2-3x-4

and shows that the degree is

2

and the constant term is

-4

. The table values locate three points, and the graph displays these intercepts visually.

Check: Substituting

x=0

into the factored form gives

(1)(-4)=-4

, matching the constant term and the table value.

Common mistakes and how to avoid them

Reading the zeros from the signs inside the factors without checking.

Correction: Set each factor equal to zero. For example,

x+1=0

gives

x=-1

.

Assuming the constant term is always a zero.

Correction: The constant term is the output when

x=0

. Zeros are inputs that make the output zero.

Treating a few table values as a complete description of the graph.

Correction: A table shows only the inputs listed. Use the equation or graph to describe additional features.

Comparing forms without saying what each one reveals.

Correction: Name the feature, then identify the representation that shows it most directly.

Lesson summary

A polynomial can have several equivalent representations.
Standard form highlights degree and constant term; factored form highlights zeros.
Tables give selected values, while graphs show visual behaviour.
Compare representations by stating what each reveals and checking that their information agrees.

Check your understanding

Question 1

For

g(x)=(x-2)(x+5)

, which representation feature makes the zeros easiest to identify?

Factored form, which shows zeros $2$ and $-5$
Standard form, which shows zeros $2$ and $-5$ directly
A table with only the input $x=0$
The constant term, which is a zero

Show answer and explanation

Factored form, which shows zeros

2

and

-5

Setting each factor equal to zero gives

x=2

and

x=-5

. Factored form displays these values most directly.

Question 2

If a polynomial is written in standard form as

h(x)=3x^2-7x+4

, what is easiest to read directly?

Its degree is $2$ and its constant term is $4$
Its zeros are $3$ and $4$
Its graph passes through $(0,7)$
Its output is $4$ for every input

Show answer and explanation

Its degree is

2

and its constant term is

4

The highest exponent is

2

, so the degree is

2

. The constant term is

4

, which is also the output when

x=0

.

Key terms

Polynomial: An expression made from numbers and whole-number powers of a variable, combined using addition, subtraction, and multiplication.
Standard form: A polynomial written with its terms ordered from the highest power to the lowest.
Factored form: A polynomial written as a product of factors.
Zero: An input that makes a polynomial's output equal to zero.
Degree: The highest exponent with a nonzero coefficient in a polynomial.
Equivalent expressions: Expressions that have the same value for every allowed input.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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