Polynomials

Q: What is the degree of a polynomial?

The highest power of the variable. 4x^3 - x + 2 has degree 3 (a cubic).

Q: What do the zeroes look like on a graph?

They are the x-values where the graph of y = p(x) crosses or touches the x-axis.

Q: What is the division algorithm for polynomials?

p(x) = g(x) \times q(x) + r(x), where the remainder r(x) is 0 or has a smaller degree than g(x).

A polynomial is an expression made of terms like $3 x^{2}, - 5 x$ and $7$ , where the powers of $x$ are whole numbers. Its degree is the highest power. A zero of a polynomial is a value of $x$ that makes it equal to 0.

Formulas

Zeroes of a quadratic ax² + bx + c

α + β = - \frac{b}{a} α β = \frac{c}{a}

Remainder theorem

p (x) \div (x - a) leaves remainder p (a)

Factor theorem

(x - a) is a factor of p (x) ⟺ p (a) = 0

Worked examples

Example 1. Find the zeroes of

x^{2} - 5 x + 6

and check the sum and product.

$x^{2} - 5 x + 6 = (x - 2) (x - 3)$ , so the zeroes are 2 and 3.
Sum $= 5 = - \frac{- 5}{1}$ ✓, product $= 6 = \frac{6}{1}$ ✓.

Answer: 2 and 3

Example 2. Find the remainder when

x^{3} - 2 x^{2} + 4 x - 1

is divided by

x - 2

By the remainder theorem, find $p (2)$ .
$p (2) = 8 - 8 + 8 - 1 = 7$ .

Answer: 7

Example 3. Is

x + 1

a factor of

x^{3} + 3 x^{2} + 3 x + 1

$x + 1 = x - (- 1)$ , so find $p (- 1)$ .
$p (- 1) = - 1 + 3 - 3 + 1 = 0$ , so yes.

Answer: Yes

Example 4. Find a quadratic polynomial whose zeroes have sum 4 and product

- 5

A quadratic is $x^{2} - (sum) x + product$ .
$x^{2} - 4 x - 5$ . Check: $(x - 5) (x + 1) = x^{2} - 4 x - 5$ , zeroes 5 and $- 1$ . ✓

Answer:

x^{2} - 4 x - 5

Common mistakes

Forgetting the minus sign in $α + β = - \frac{b}{a}$ .
For a divisor $x + a$ , putting $x = a$ . The value to use is $x = - a$ .
Calling an expression with $x$ or $\frac{1}{x}$ a polynomial. Powers of $x$ must be whole numbers.

Practice questions with answers

1. What is the degree of

4 x^{5} - 2 x^{3} + x - 7

Show answer

Answer: A) 5

Degree = highest power of variable. Here it's 5 (from

x^{5}

2. What is the zero of the polynomial

p (x) = x + 7

A-14
B-7
C-8
D-6

Show answer

Answer: B) -7

Set

p (x) = 0 : x + 7 = 0

, so

x = - 7

3. By the Remainder Theorem, the remainder when a polynomial p

(x)

is divided by

(x - a)

equals which value?

Ap(a)
B $p (- a)$
Cp(0)
Dp(1)

Show answer

Answer: A) p(a)

The Remainder Theorem states that if p

(x)

is divided by

(x - a)

, the remainder is p(a).

4. By the Factor Theorem,

(x - a)

is a factor of a polynomial p

(x)

if and only if which condition holds?

A $p (a) = 0$
B $p (0) = a$
C $p (a) = 1$
D $p (- a) = 0$

Show answer

Answer: A)

p (a) = 0

Factor Theorem:

(x - a)

is a factor of p

(x)

exactly when the remainder p(a) equals 0.

5. A polynomial of degree 1 is called a _____ polynomial.

Alinear
Bquadratic
Ccubic
Dconstant

Show answer

Answer: A) linear

A polynomial of degree 1 (like ax + b, a not 0) is called a linear polynomial.

6. Sum of zeros of

3 x^{2} - 6 x + 2

A2
B−2
C $\frac{2}{3}$
D−6

Show answer

Answer: A) 2

Sum = −b/a = −(−6)/3 = 2.

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Frequently asked questions

What is the degree of a polynomial?

The highest power of the variable. $4 x^{3} - x + 2$ has degree 3 (a cubic).

How many zeroes can a polynomial have?

At most as many as its degree: a quadratic has at most 2, a cubic at most 3.

What do the zeroes look like on a graph?

They are the $x$ -values where the graph of $y = p (x)$ crosses or touches the $x$ -axis.

What is the division algorithm for polynomials?

$p (x) = g (x) \times q (x) + r (x)$ , where the remainder $r (x)$ is 0 or has a smaller degree than $g (x)$ .

Polynomials

Formulas

Worked examples

Common mistakes

Practice questions with answers

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Frequently asked questions

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