Welcome! Today we're exploring Zeros in Arithmetic Progression and Cubic Symmetric Functions — a powerful pattern that shows up again and again in cubic problems.

A favourite pattern in cubic problems: the three zeros are in arithmetic progression (AP).

When zeros follow a pattern, the algebra becomes surprisingly clean.

The smart notation we'll use:

Zero	Expression
First	$(a - d)$
Middle	$a$
Third	$(a + d)$

This choice makes the sum formula give $3a$ key result — immediately revealing the middle zero.

Then the sum of pairwise products gives you (find d) $d$ .

By the end, you will master this two-step technique:

Use sum of zeros → find $a$
Use pairwise products → find $d$

1. The AP zeros notation and sum simplification

Let's look at a clever trick for working with cubic polynomials!

When three zeros of a cubic are in arithmetic progression (AP), we can write them as:

Number line showing three points labeled (a-d), a, and (a+d) equally spaced, with arrows indicating equal distance d between consecutive points

(a-d), \quad a, \quad (a+d)

This notation creates a beautiful simplification: when you add them up, the $d$ -terms cancel out!

\text{Sum} = (a-d) + a + (a+d) = 3a

This means the sum formula alone determines the middle zero $a$ .

✍️ Question

Your turn! 🤔

The zeros of a cubic polynomial are in AP: $(a-d)$ , $a$ , $(a+d)$ .

If the sum of zeros is 6, what is $a$ find this?

The APNumbers are equally spaced, same gap between each pair notation for zeros:

When three zeros are in Arithmetic ProgressionSame gap between each pair of numbers, we write them as $(a-d)$ Standard way to write three AP terms, $a$ middleAutomatically builds in equal spacing from the middle, and $(a+d)$ Standard way to write three AP terms

Number line showing three equally spaced points labeled (a-d), a, and (a+d), with arrows showing distance d between consecutive points

Here:

$a$ is the middle valueEverything is measured from it
$d$ is the common differenceHow far apart each term is from the next

✍️ MCQ

Choose one

In the AP notation

(a-d)

,

a

,

(a+d)

, which term represents the middle zero?

✓

Finding the sum:

Let's add these three zeros:

$(a-d) + a + (a+d)$

Expanding this:

$(a - d) + a + (a + d) =$ $a - d + a + a + d$ $=$ $3a$ sum(The sum simplifies to 3a, three times the middle term)

Notice what happened? The $d$ terms cancel out!The d terms cancel, leaving only the middle value So the sum of zeros depends only on $a$ , not on $d$ .

✍️ MCQ

Choose one

In our problem, the sum of zeros is

6

. Using the result that the sum equals

3a

, what is the value of

a

?

✓

Applying this to our problem:

If the sum of zeros $= 6$ :

3a = 6

a = 2

answer(Sum divided by 3 gives the middle value)

So $a = 2$ This is the middle zero we found is the middle zero. We don't need to know $d$ to find the middle value — the sum alone determines itNo need to find d or the other zeros.

✍️ MCQ

Choose one

If the sum of three AP zeros is

15

, what is the middle zero

a

?

✓

2. Using pairwise products to find d

📋 Given Info

Let's continue with our AP zeros problem.

With a now known, the sum of pairwise products gives us an equation in d (or b in our notation).

Key insight: When you expand the pairwise products for zeros $(a - b)$ , $a$ , $(a + b)$ , the cross terms cancel beautifully, leaving:

3a^2 - b^2

key formula

✍️ Question

Your Turn 📝

The zeros of $x^{3} - 3 x^{2} + x + 1$ are $(a - b)$ , $a$ , $(a + b)$ .

Given that $a = 1$ (from the sum), find $b$ using the sum of pairwise products relationship.

Hint: Remember that for a cubic $x^{3} + p x^{2} + q x + r$ , the sum of pairwise products of zeros equals $q$ .

For zeros $(a - b)$ , $a$ , $(a + b)$ , let's expand the sum of pairwise productsMultiply every pair of zeros and add them up:

$(a - b) (a) + (a) (a + b) + (a + b) (a - b)$

Expanding each term:

$= a^{2} - a b + a^{2} + a b + a^{2} - b^{2}$

$=$ $3a^2$ The clean form we'll use in problems $-$ $b^2$ The clean form we'll use in problems

✍️ MCQ

Choose one

Why did the

ab

terms cancel in the expansion?

✓

Notice how the $ab$ terms cancelcancel!Cross terms always vanish due to symmetry — the AP structureCreates automatic cancellation of cross terms creates this simplification automatically!

This is exactly like what happened with the sum of zeros — the symmetric spacing around the middle valueZeros are evenly spaced around center makes cross terms vanishThey cancel because of symmetric spacing.

For $x^{3} - 3 x^{2} + x + 1$ : the sum of pairwise productsAlpha beta plus beta gamma plus gamma alpha equals $\frac{c}{a_{c o e f f}} =$ $\frac{1}{1} = 1$ So this sum equals 1.

So we have: $3a^2 - b^2 = 1$ (Connects our AP parameters to the polynomial)

Substituting $a = 1$ from sumFrom our earlier work: $3 (1)^{2} - b^{2} = 1$

✍️ MCQ

Choose one

We have

3(1)^2 - b^2 = 1

. What is

b^2

?

✓

This gives us $b^2 = 2$ The calculation gives us this value, so $b = \sqrt{2}$ That's our common difference.

So the three zeros are: $(1 - \sqrt{2})$ First zero in the progression, $1$ centerThe middle value, and $(1 + \sqrt{2})$ Last zero in the progression.

The three zeros are: $1 - \sqrt{2}$ First zero, symmetric with the third, $1$ The center of the arithmetic progression, $1 + \sqrt{2}$ Third zero, symmetric with the first.

Number line showing three points at 1-sqrt(2), 1, and 1+sqrt(2), with equal spacing marked to show arithmetic progression

Note that both values $b = \sqrt{2}$ One value of b and $b = -\sqrt{2}$ The other value of b give the same three zerosSymmetric zeros mean the set doesn't change — just in different order.

✍️ MCQ

Choose one

If the zeros of a cubic are in AP as

(a-d)

,

a

,

(a+d)

, and the sum of zeros is

9

, what is the middle zero

a

?

✓