MHF4U · Unit 1 test review

Polynomial Functions: Test Review

Everything for the Unit 1 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Polynomial Functions and Their Shapes

Standard form: $f (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ with $a_{n} \neq = 0$ and all exponents whole numbers.
End behaviour depends only on the leading term: even degree means both ends agree, odd degree means they oppose; the sign of $a_{n}$ says which way.
A degree- $n$ polynomial has at most $n$ zeros and at most $n - 1$ turning points.
Domain is always ${x \in R}$ ; the range is ${y \in R}$ exactly when the degree is odd.
For a degree- $n$ polynomial with $x$ -values spaced by $1$ , the $n$ th finite differences are constant and equal $a \cdot n!$ .
The constant term $a_{0}$ equals $f (0)$ , the $y$ -intercept.

Zeros, Multiplicity and Sketching

A factor $(x - r)$ gives the zero $x = r$ ; a factor $(a x - b)$ gives the zero $x = \frac{b}{a}$ .
In $a (x - r_{1})^{m_{1}} \dots (x - r_{k})^{m_{k}}$ , the multiplicities add to the degree.
Odd multiplicity means the graph crosses the axis; even multiplicity means it touches and turns back.
Multiplicity $3$ or higher flattens the graph against the axis near that zero.
The $y$ -intercept is $f (0)$ — substitute $x = 0$ into the factored form directly.
Zeros alone determine the curve only up to a vertical stretch $a$ ; one extra point fixes $a$ .

The Remainder and Factor Theorems

Division statement: $P (x) = D (x) Q (x) + R (x)$ , with $de g R < de g D$ .
Remainder theorem: dividing $P (x)$ by $x - a$ leaves remainder $P (a)$ ; dividing by $a x - b$ leaves $P (\frac{b}{a})$ .
Factor theorem: $x - a$ is a factor of $P (x)$ if and only if $P (a) = 0$ .
Integer zero theorem: an integer zero of a polynomial with integer coefficients must divide the constant term.
Rational zero theorem: a rational zero $\frac{p}{q}$ has $p ∣ a_{0}$ and $q ∣ a_{n}$ .
Synthetic division: bring down, multiply by $a$ , add. Last entry is the remainder; the rest are the quotient's coefficients.

Families of Polynomials and Polynomial Models

The family with zeros $r_{1}, \dots, r_{n}$ is $y = k (x - r_{1}) \dots (x - r_{n})$ , $k \neq = 0$ ; substitute one non-zero point to find $k$ .
Irrational zeros $a \pm b$ come as a pair whose factors multiply to $(x - a)^{2} - b$ .
Even: $f (- x) = f (x)$ , symmetric in the $y$ -axis. Odd: $f (- x) = - f (x)$ , symmetric about the origin.
A polynomial with only even powers (constants included) is even; only odd powers, odd; a mixture, neither.
Open box from an $L \times W$ sheet: $V (x) = x (L - 2 x) (W - 2 x)$ with domain $0 < x < \frac{W}{2}$ (the shorter side).
Solve a target-value model by rearranging to $P (x) = 0$ , using the factor theorem, then rejecting roots outside the domain.

Polynomial Inequalities

Always rearrange to compare with $0$ before factoring. Never divide both sides by an expression containing $x$ .
The sign of a polynomial can change only at its zeros, so the zeros cut the number line into test intervals.
The sign flips across a zero of odd multiplicity and does not flip across a zero of even multiplicity.
Strict inequalities ( $<$ , $>$ ) exclude the zeros and use round brackets; non-strict ( $\leq$ , $\geq$ ) include them and use square brackets.
$\infty$ and $- \infty$ always take round brackets: $[2, \infty)$ , never $[2, \infty]$ .
Separate pieces of a solution are joined with a union symbol, as in $(- 2, 0) \cup (2, \infty)$ .

Where marks get lost

Calling $y = x$ , $y = \frac{2}{x}$ or $y = 3^{x}$ polynomials. Whole-number exponents only — no roots, no variables in denominators, no variables in exponents. (Polynomial Functions and Their Shapes)
Grabbing the first coefficient written down instead of the leading one. In $f (x) = 7 - 2 x^{3} + x^{2}$ the leading coefficient is $- 2$ , not $7$ — rearrange into standard form first. (Polynomial Functions and Their Shapes)
Saying a quartic "has 3 turning points". It has at most $3$ ; it could have $1$ . The same goes for zeros: at most $n$ , possibly fewer. (Polynomial Functions and Their Shapes)
Computing finite differences from a table whose $x$ -values are not equally spaced. Check the spacing before you subtract anything. (Polynomial Functions and Their Shapes)
Claiming the range of every polynomial is all real numbers. That is only true for odd degree; $y = x^{4} - 3$ has range ${y \geq - 3}$ . (Polynomial Functions and Their Shapes)
Reading the factor $(x + 2)$ as the zero $x = 2$ . Set the bracket equal to zero: $x + 2 = 0$ gives $x = - 2$ . The sign always flips. (Zeros, Multiplicity and Sketching)
Assuming every zero is a crossing. A squared factor such as $(x - 1)^{2}$ makes the graph touch and turn back, so the curve keeps the same sign on both sides of it. (Zeros, Multiplicity and Sketching)
Writing an equation from a graph as $y = (x + 1)^{2} (x - 2)$ and stopping. Without solving for the stretch factor $a$ , the curve will have the right intercepts but the wrong height everywhere else. (Zeros, Multiplicity and Sketching)
Counting distinct zeros as the degree. For $f (x) = (x - 1)^{2} (x + 3)^{3}$ there are only two distinct zeros, but the degree is $2 + 3 = 5$ . (Zeros, Multiplicity and Sketching)
Drawing a sharp corner or a break at a zero. Polynomial graphs are always smooth and continuous — a corner means you have drawn something else. (Zeros, Multiplicity and Sketching)
Substituting the wrong sign. For the divisor $x + 3$ you test $x = - 3$ , because $- 3$ is what makes the bracket zero. The rule is "the value that zeroes the divisor", never "the number you see". (The Remainder and Factor Theorems)
Forgetting placeholder zeros in synthetic division. Dividing $x^{3} - 7 x + 6$ means using the coefficients $1, 0, - 7, 6$ — the missing $x^{2}$ term must appear as a $0$ . (The Remainder and Factor Theorems)
Testing only positive candidates. Negative zeros are just as common; work through $\pm 1$ , $\pm 2$ and so on in pairs. (The Remainder and Factor Theorems)
For a divisor like $2 x + 1$ , testing $x = - 2$ instead of $x = - \frac{1}{2}$ . Solve $2 x + 1 = 0$ properly before substituting. (The Remainder and Factor Theorems)
Stopping after one factor. $P (x) = (x - 2) (2 x^{2} + 5 x + 2)$ is not fully factored — the quadratic still breaks into $(2 x + 1) (x + 2)$ . (The Remainder and Factor Theorems)
Writing the family with the zero's sign instead of the opposite: a zero of $- 2$ gives the factor $(x + 2)$ , not $(x - 2)$ . (Families of Polynomials and Polynomial Models)
Forgetting $k$ entirely, so the equation has the right zeros but does not pass through the given point. (Families of Polynomials and Polynomial Models)
Calling $p (x) = x^{3} + 1$ odd because its leading term is odd. The constant is an even power, so it is neither. (Families of Polynomials and Polynomial Models)
Reporting every root of a model equation, including ones such as $x \approx 17.07$ cm that lie outside the physical domain. (Families of Polynomials and Polynomial Models)
Using $0 < x < 15$ for a box from a $30 \times 20$ sheet. The SHORTER side limits the cut, so the domain is $0 < x < 10$ . (Families of Polynomials and Polynomial Models)
Dividing both sides by $x$ (or by any factor containing $x$ ). You lose solutions and may need to flip the sign — move everything to one side instead. (Polynomial Inequalities)
Factoring before rearranging. Writing $x^{3} > 4 x$ as $x (x^{2}) > 4 x$ achieves nothing; you need $x^{3} - 4 x > 0$ first. (Polynomial Inequalities)
Including the endpoints on a strict inequality. For $f (x) > 0$ the zeros make $f (x) = 0$ , which is not greater than $0$ , so they are excluded. (Polynomial Inequalities)
Assuming the signs alternate across every zero. At a zero of even multiplicity the sign stays the same, and skipping that check flips half your answer. (Polynomial Inequalities)
Writing $[2, \infty]$ or joining separate intervals with "and". Infinity always gets a round bracket, and disjoint pieces are joined with a union. (Polynomial Inequalities)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Polynomial Functions and Their Shapes
1
Which of the following is NOT a polynomial function?
Polynomial Functions and Their Shapes
2
What is the greatest possible number of turning points on the graph of a degree-7 polynomial function?
Polynomial Functions and Their Shapes
3
State the leading coefficient of $f (x) = 7 - 2 x^{3} + x^{2}$ .
Polynomial Functions and Their Shapes
4
What is the constant third difference for $f (x) = - 4 x^{3} + x$ , using $x$ -values that step by $1$ ?
Polynomial Functions and Their Shapes
5
Match each degree-and-sign combination to its end behaviour.
Even degree, a > 0
Even degree, a < 0
Odd degree, a > 0
Odd degree, a < 0
Zeros, Multiplicity and Sketching
6
What are the zeros of $f (x) = (2 x - 3) (x + 5)^{2}$ ?
Zeros, Multiplicity and Sketching
7
State the degree of $y = 3 (x - 1)^{3} (x + 2)^{2} (x - 5)$ .
Zeros, Multiplicity and Sketching
8
In one phrase, describe what the graph does at a zero of even multiplicity.
Zeros, Multiplicity and Sketching
9
What is the maximum possible number of turning points on the graph of $y = (x - 1)^{2} (x + 4)^{3}$ ?
Zeros, Multiplicity and Sketching
10
Match each multiplicity to the behaviour of the graph at that zero.
Multiplicity 1
Multiplicity 2
Multiplicity 3
Multiplicity 4
The Remainder and Factor Theorems
11
What is the remainder when $P (x) = x^{3} - 2 x^{2} + 5 x - 1$ is divided by $x - 3$ ?
The Remainder and Factor Theorems
12
For what value of $k$ is $x - 2$ a factor of $P (x) = x^{3} + k x^{2} - 4 x + 8$ ?
The Remainder and Factor Theorems
13
According to the remainder theorem, what value of $x$ do you substitute to find the remainder when $P (x)$ is divided by $x + 4$ ?
The Remainder and Factor Theorems
14
When $x^{3} - 4 x^{2} + x + 6$ is divided by $x - 3$ , what is the constant term of the quotient?
The Remainder and Factor Theorems
15
Match each divisor to the value you substitute to find the remainder.
x − 5
x + 2
2x − 3
3x + 1
Families of Polynomials and Polynomial Models
16
The cubic $y = k (x - 1) (x + 3) (x - 2)$ passes through $(0, 12)$ . Find $k$ .
Families of Polynomials and Polynomial Models
17
Classify $f (x) = 3 x^{5} - x^{3} + 2 x$ .
Families of Polynomials and Polynomial Models
18
Match each function to its symmetry.
$x^{4} - 5 x^{2} + 4$
$2 x^{3} - 5 x$
$x^{3} + 1$
$x^{3} + x^{2}$
Families of Polynomials and Polynomial Models
19
$V (x) = 1000$ for the $30 \times 20$ box has roots $5$ and $10 \pm 50$ . Give the other valid cut, to 2 decimal places.
cm
Families of Polynomials and Polynomial Models
20
Which polynomial belongs to the same family as $y = (x + 2) (x - 1) (x - 3)$ ?
Polynomial Inequalities
21
What is the solution to $(x - 3) (x + 1) < 0$ ?
Polynomial Inequalities
22
A polynomial has 4 distinct real zeros. How many intervals does its sign chart have?
Polynomial Inequalities
23
What is the smallest integer in the solution set of $(x - 1) (x - 4) \leq 0$ ?
Polynomial Inequalities
24
How many separate intervals make up the solution set of $(x + 3)^{2} (x - 1) > 0$ ?
Polynomial Inequalities
25
Match each inequality to its interval notation.
x > 2
x ≥ 2
x < 2
x ≤ 2

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…