Polynomials: Formulas

Class 10

Sum of the zeroes of a quadratic

S = - \frac{b}{a}

What each letter means
$S$	the sum of the two zeroes, α + β, of ax² + bx + c
$a$	the coefficient of x² (not 0)
$b$	the coefficient of x

The two zeroes of ax² + bx + c always add up to −b ÷ a, so you can know their sum without finding them.

Why it works

If α and β are the zeroes, the polynomial is a(x − α)(x − β) = ax² − a(α + β)x + aαβ. Matching the x terms, b = −a(α + β), so α + β = −b ÷ a.

When to use it

Checking the zeroes you found, finding the second zero from the first, writing a quadratic from the sum and product of its zeroes, or finding the midpoint of where a graph crosses the x-axis.

How to use it

Write the quadratic as ax² + bx + c and read off a and b (with their signs).
Change the sign of b and divide by a.

To remember: Sum is minus b over a.

Other forms

Same formula, another formFor a cubic ax³ + bx² + cx + d with zeroes α, β, γ, the sum α + β + γ is also −b ÷ a. For 3x³ − 5x² − 11x − 3 (zeroes 3, −1 and −1/3) it is 5/3.
Special case $S = 0$ When there is no x term (b = 0): the zeroes are opposites, like √3 and −√3 for x² − 3.

Worked examples

Story

A rectangle's length and breadth (in cm) are the two zeroes of x² − 17x + 60. Find its perimeter without finding the sides.

Length + breadth = the sum of the zeroes = −(−17) ÷ 1 = 17 cm.
$2 \times 17 = 34$
The perimeter is 34 cm. Check: the zeroes are 12 and 5, and 2(12 + 5) = 34.

Answer: 34 cm

Picture

The graph of y = x² − 4x − 5 crosses the x-axis at two points. Without drawing it, find the x-coordinate of the point halfway between them.

The crossing points are the zeroes, and their sum is −(−4) ÷ 1 = 4.
The halfway point is the average of the zeroes: 4 ÷ 2 = 2.
Check: the zeroes are −1 and 5, and (−1 + 5) ÷ 2 = 2. The graph is symmetric about the line x = 2.

Answer: x = 2

Direct

Find the sum of the zeroes of x² + 7x + 10, and check it by finding the zeroes.

$S = - \frac{7}{1} = - 7$
x² + 7x + 10 = (x + 2)(x + 5), so the zeroes are −2 and −5, and −2 + (−5) = −7.

Answer: −7

Reverse

Find a quadratic polynomial whose zeroes add to −3 and multiply to 2.

Try it first, then show the working

A quadratic with zeroes α and β is x² − (α + β)x + αβ.
$x^{2} - (- 3) x + 2 = x^{2} + 3 x + 2$
Check: x² + 3x + 2 = (x + 1)(x + 2), with zeroes −1 and −2: they add to −3 and multiply to 2.

Answer: x² + 3x + 2

Exam

One zero of 2x² − 11x + 15 is 3. Find the other zero without solving the equation.

Try it first, then show the working

The zeroes add to −(−11) ÷ 2 = 11/2.
$\frac{11}{2} - 3 = \frac{5}{2}$
Check: 2 × (5/2)² − 11 × (5/2) + 15 = 12.5 − 27.5 + 15 = 0.

Answer: 5/2

Common mistake: Forgetting the minus sign (x² + 7x + 10 has zeroes −2 and −5, which add to −7, not 7), or reading b from a polynomial that is not in the order ax² + bx + c.

Class 10

Product of the zeroes of a quadratic

P = \frac{c}{a}

What each letter means
$P$	the product of the two zeroes, αβ, of ax² + bx + c
$a$	the coefficient of x² (not 0)
$c$	the constant term (the number with no x)

The two zeroes of ax² + bx + c always multiply to c ÷ a.

Why it works

If α and β are the zeroes, the polynomial is a(x − α)(x − β) = ax² − a(α + β)x + aαβ. Matching the constant terms, c = aαβ, so αβ = c ÷ a.

When to use it

Checking zeroes, finding the second zero from the first, writing a quadratic from its zeroes, or when the zeroes stand for two lengths whose product (an area) you need.

How to use it

Write the quadratic as ax² + bx + c and read off a and c (with their signs).
Divide c by a.

To remember: Product is c over a.

Other forms

Same formula, another formFor a cubic ax³ + bx² + cx + d with zeroes α, β, γ: αβγ = −d ÷ a, and αβ + βγ + γα = c ÷ a. For 3x³ − 5x² − 11x − 3 (zeroes 3, −1, −1/3): the product is 1 and the pair products add to −11/3.
Special case $P = 0$ When there is no constant term (c = 0): then 0 is one of the zeroes.

Worked examples

Story

A rectangle's length and breadth (in cm) are the zeroes of x² − 17x + 60. Find its area without finding the sides.

Area = length × breadth = the product of the zeroes.
$P = \frac{60}{1} = 60$
The area is 60 cm² (the sides are 12 cm and 5 cm).

Answer: 60

Picture

Both zeroes of x² − 6x + 9 are the side of the same square (the zeroes are equal). Find the area of the square.

The zeroes are equal, so each is the side s, and s × s = the product of the zeroes = 9 ÷ 1.
The area is 9 square units (the side is 3).

Answer: 9 square units

Direct

Find the product of the zeroes of x² + 7x + 10, and check it.

$P = \frac{10}{1} = 10$
The zeroes are −2 and −5, and (−2) × (−5) = 10.

Answer: 10

Reverse

One zero of 3x² − 7x + 2 is 2. Use the product of the zeroes to find the other.

Try it first, then show the working

The zeroes multiply to 2 ÷ 3.
$\frac{2}{3} \div 2 = \frac{1}{3}$
Check: 3 × (1/3)² − 7 × (1/3) + 2 = 1/3 − 7/3 + 2 = 0.

Answer: 1/3

Exam

The zeroes of x² − 3 are √3 and −√3. Verify the relations between the zeroes and the coefficients.

Try it first, then show the working

Here a = 1, b = 0 and c = −3.
Sum: √3 + (−√3) = 0, and −b ÷ a = 0.
Product: √3 × (−√3) = −3, and c ÷ a = −3 ÷ 1 = −3. Both relations hold.

Answer: Sum 0 = −b/a and product −3 = c/a

Common mistake: Dividing a by c instead of c by a, or dropping the sign of c (x² − 3 has zeroes √3 and −√3, whose product is −3).