Unit notes and practice questions

Grade 10 Mathematics unit 2: Polynomial Functions

Everything here is for unit 2 of the Grade 10 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

5
Cards: 10
Practice questions on unit 2: 65–115
The textbook pages for unit 2

What unit 2 covers

1.Introduction of polynomial functions
2.Operations on polynomial functions
3.Theorem on polynomial functions
4.Zeros of polynomial functions
5.Graphs of polynomial functions
6.Applications

Short notes for unit 2

The cards filed under this chapter on the Grade 10 Mathematics page. Open one at a time.

End Behavior of Polynomial GraphsReference tableOdd · Even

Degree	Leading Coefficient	Left End Behavior	Right End Behavior
Odd	$a_n > 0$	Downwards	Upwards
Odd	$a_n < 0$	Upwards	Downwards
Even	$a_n > 0$	Upwards	Upwards
Even	$a_n < 0$	Downwards	Downwards

When you use it

Use to determine how the ends of a polynomial graph behave far to the left and far to the right based on degree and leading coefficient.

Watch out

Remember that polynomial graphs are always smooth and continuous, having no sharp corners, jumps, or holes.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Polynomial Division TheoremFormula

f(x) = d(x)q(x) + r(x)

f(x) = d(x)q(x) + r(x)

$f(x)$: dividend polynomial $dimensionless$
$d(x)$: non zero divisor polynomial $dimensionless$
$q(x)$: quotient polynomial $dimensionless$
$r(x)$: remainder polynomial whose degree is less than d(x) or is zero $dimensionless$

When you use it

Use to express a polynomial in terms of a divisor, quotient, and remainder during polynomial long division.

Watch out

Stop the division process only when the remainder is zero or its degree is strictly less than the divisor degree.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Rational Root TestMethod

When you use it

Use to find all potential rational zeros p/q for a polynomial function with integer coefficients.

Watch out

The numerator p must be a factor of the constant term a_0, and the denominator q must be a factor of the leading coefficient a_n, not the reverse.

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Remainder and Factor TheoremsRule

When you use it

Use to find the remainder when f(x) is divided by x - c, or to confirm that x - c is a factor by checking if f(c) = 0.

Watch out

When dividing by x + c, evaluate f(-c) because the standard divisor form is x - (-c).

Drafted from Grade 10 Mathematics, pages 66-116, then checked twice before it went up

Special products and factoring identitiesFormula

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

$x$: the variable, or any expression standing in its place
$a$: a constant, or an expression, the same one everywhere in the identity
$b$: the second constant in the monic quadratic rule
$n$: a positive whole number, in the even powers rule

Expression	Its factorised form
$x^{2} - a^{2}$	$(x+a)(x-a)$
$x^{2} + 2ax + a^{2}$	$(x+a)^{2}$
$x^{2} - 2ax + a^{2}$	$(x-a)^{2}$
$x^{2} + (a+b)x + ab$	$(x+a)(x+b)$
$x^{3} + 3ax^{2} + 3a^{2}x + a^{3}$	$(x+a)^{3}$
$x^{3} + a^{3}$	$(x+a)(x^{2} - ax + a^{2})$
$x^{3} - a^{3}$	$(x-a)(x^{2} + ax + a^{2})$
$x^{2n} - a^{2n}$	$(x^{n} - a^{n})(x^{n} + a^{n})$

When you use it

Reach for these when an expression matches one of the patterns and you need it written as a product: solving a polynomial equation, cancelling a fraction, or simplifying before you differentiate. Read the expression against the list first, because a pattern that matches saves the whole trial and error search. The even powers line is the difference of squares with $x^{n}$ in place of $x$ , and each of its two factors will often break down further.

Watch out

The sign inside the trinomial is the opposite of the sign in the bracket. Students copy the binomial's sign across and write $x^{3} + a^{3} = (x+a)(x^{2} + ax + a^{2})$ , which is wrong: put $x = 2$ and $a = 3$ into it and you get 95 instead of 35. The last term is $+a^{2}$ in both cubes. Two more that cost marks on the same page. A sum of squares $x^{2} + a^{2}$ has no factorisation over the real numbers, so it is never $(x+a)^{2}$ . And the $x^{2} + (a+b)x + ab$ rule needs the coefficient of $x^{2}$ to be 1, so it does nothing for $2x^{2} + 5x + 3$ until you take the 2 out or split the middle term.

Practice questions on unit 2

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

What is the degree of the polynomial function h(x) = -5x⁴ + 8x³ + 2x² - 3x + 7?

A.3
B.4
C.7
D.-5

Show the answer and why

Answer: B 4

Why

The degree is the exponent of the highest power of x. Here the highest power is x⁴, so the degree is 4. The number -5 is the leading coefficient, not the degree.

Question 2

Which of the following is NOT a polynomial function?

A.f(x) = 2x2 - 9x3 + (5/2)x
B.g(x) = (2 - x)(2 + x)
C.k(y) = √y
D.f(x) = 2x3 + 5x - 1

Show the answer and why

Answer: C k(y) = √y

Why

A polynomial function must be written with only non-negative whole number powers of the variable. In k(y) = √y, the variable y is inside a radical sign, so it cannot be written in polynomial form. The other three can all be written in the polynomial form.

Question 3

What is the domain of every polynomial function?

A.All real numbers
B.All positive numbers only
C.All integers only
D.Only x greater than or equal to 0

Show the answer and why

Answer: A All real numbers

Why

The chapter says that the domain of a polynomial function is all real numbers. You can put any real number into a polynomial.

Question 4

What is the constant term of the polynomial function f(x) = 12x³ - 9x² + 3x + 4?

A.12
B.-9
C.3
D.4

Show the answer and why

Answer: D 4

Why

The constant term is the term that has no x in it. In this function it is +4. The number 12 is the leading coefficient, and it is not the constant term.

Question 5

Let f(x) = 2x⁴ + 5x³ + x² + 9x + 4 and g(x) = -3x³ + 5x² - 5x - 9. Find (f + g)(x).

A.2x4 + 2x3 + 6x2 + 4x - 5
B.2x4 + 8x3 + 6x2 + 4x - 5
C.2x4 + 2x3 + 6x2 + 14x - 5
D.2x4 + 2x3 - 4x2 + 4x - 5

Show the answer and why

Answer: A 2x4 + 2x3 + 6x2 + 4x - 5

Why

To add polynomials, add the coefficients of like terms.

x⁴: 2. x³: 5 + (-3) = 2. x²: 1 + 5 = 6. x: 9 + (-5) = 4. Constant: 4 + (-9) = -5.

So (f + g)(x) = 2x⁴ + 2x³ + 6x² + 4x - 5.

Question 6

Let f(x) = 2x + 3 and g(x) = 3x² - 5x + 6. Find (f · g)(x).

A.6x3 - x2 - 3x - 18
B.6x3 - 10x2 + 18
C.6x3 - x2 - 3x + 18
D.6x3 + x2 - 3x + 18

Show the answer and why

Answer: C 6x3 - x2 - 3x + 18

Why

Multiply each term of one polynomial by every term of the other, then add like terms.

2x(3x² - 5x + 6) = 6x³ - 10x² + 12x.

3(3x² - 5x + 6) = 9x² - 15x + 18.

Adding: 6x³ - x² - 3x + 18.

Question 7

Let f(x) = 2x³ + 5x² + 3x + 2. Using the remainder theorem, what is the remainder when f(x) is divided by x + 1?

A.12
B.2
C.-2
D.0

Show the answer and why

Answer: B 2

Why

x + 1 = x - (-1), so c = -1. The remainder is f(-1).

f(-1) = 2(-1)³ + 5(-1)² + 3(-1) + 2 = -2 + 5 - 3 + 2 = 2.

The remainder is 2. (The value 12 is f(1), not f(-1).)

Question 8

For the function f(x) = x² + 2x - 3, the chapter rewrites it as f(x) = -4 + (x + 1)². What is the turning point of the graph?

A.(1, 0)
B.(-1, -4)
C.(-2, -3)
D.(2, 5)

Show the answer and why

Answer: B (-1, -4)

Why

Since (x + 1)² is never negative, the smallest value of f is -4. It happens when (x + 1)² = 0, which means x = -1. The turning point (vertex) is (-1, -4).

Question 9

Find the quotient q(x) when 4x³ + 4x² - x + 4 is divided by 2x - 1.