Welcome! Today we're exploring The Zero Product Property and the Factor Theorem — the single most important principle for finding zeros of polynomials.

It starts with a simple fact about numbers:

If two things multiply to give 0, at least one of them must be 0.

From this, the factor theorem builds a bridge:

Connection	Meaning
Zero → Factor	Every zero corresponds to a factor
Factor → Zero	Every factor reveals a zero

Understanding this bridge is essential because factoring is the primary method for finding zeros throughout this chapter.

1. The zero product property

🎯 The Zero Product Property

Everything starts with a basic fact about multiplication:

If a product equals 0, then at least one of the factors must be 0.

This is the foundation on which the entire factoring approach is built.

📋 Given Info

The Rule:

If $A \times B = 0$ , then $A = 0$ or $B = 0$ (or both).

Think about it:

$3 \times 7$ cannot be $0$
$(-5) \times 2$ cannot be $0$

For a product to be $0$ , at least one factor must be $0$ .

✍️ Question

Your Turn ✏️

If $p (x) = (x + 4) (2 x - 3) = 0$ , what are the zeros of $p (x)$ ?

Show the solving step for each factor.

The Zero Product PropertyYour main tool for factored equations: If $A \times B = 0$ When a product equals zero, then $A = 0$ or $B = 0$ One of the factors must be zero (or both).

This is the foundation — when a product equals zero, at least one of the factors must be zeroRequired for a product to equal zero.

Think about it: $3 \times 7 = 21$ , not zero. $(- 5) \times 2 = - 10$ , not zero. The only way to get zeroNo other combination works is if at least one factor is already zerokeyThis is the key requirement!

✍️ MCQ

Choose one

If

(x - 5)(x + 2) = 0

, which of the following must be true?

✓

For $(x + 4) (2 x - 3) = 0$ :

We have a product of two factorsThis is the structure that triggers the property equal to zerokeyWhen this happens, Zero Product Property kicks in.

✍️ MCQ

Choose one

According to the Zero Product Property, if

(x + 4)(2x - 3) = 0

, what must be true?

✓

Case 1: Either $x + 4 = 0$

Solving: $x = -4$

Case 2: Or $2x - 3 = 0$

Solving: $2x = 3$ , so $x = \frac{3}{2}$

✍️ FIB

Fill in the blank

We found

x = -4

from the first factor. What value of

x

did we get from solving

2x - 3 = 0

?

✓

The zeros of $p (x)$ are: $x = -4$ zero 1First factor gives first zero and $x = \frac{3}{2}$ zero 2Second factor gives second zero

⚠️ A common trap:Most common error in factorization Seeing $+ 4$ in $(x + 4)$ and writing $x = +4$ wrong!Appears frequently on tests.

This is one of the most frequent mistakes students make!

✍️ MCQ

Choose one

If

(x + 4) = 0

, is the solution

x = +4

or

x = -4

?

✓

But $x + 4 = 0$ means $x = -4$ Set equal to zero and solve. The sign FLIPSSign changes when solving.

📌 Always solve the equationDon't just copy the number from the bracket rather than just reading off the number from the factor.

Quick check: Substitute $x = 4$ into $(x + 4)$ :

$4 + 4 =$ $8 \neq 0$

Now substitute $x = - 4$ :

$- 4 + 4 = 0 ✓$

The zero is $x = -4$ answerThe zero is negative 4, not positive 4, not $x = +4$ Don't just read off the number you see.

✍️ MCQ

Choose one

If

(x - 7) = 0

, what is

x

?

✓

2. The factor theorem as a two-way bridge

The zero product property tells us how to get zeros from factors. The factor theorem goes further: it says the bridge works in both directions.

If $α$ is a zero, then $(x - α)$ is a factor, and vice versa.

Direction	What it means
Zero → Factor	If $α$ is a zero of $p (x)$ , then $(x - α)$ is a factor
Factor → Zero	If $(x - α)$ is a factor of $p (x)$ , then $α$ is a zero

The Factor Theorem states:

A real number $α$ is a zero of $p (x)$ if and only if $(x - α)$ is a factor of $p (x)$ .

Notice the "if and only if" — both directions hold.

✍️ Question

Question 🤔

If $(x + 3)$ is a factor of a polynomial $p (x)$ , what is the corresponding zero?

Be careful with the sign.

The factor theorem says: (x - α)If alpha is a zero, this is the factor is a factor when αThe zero value that makes the polynomial equal zero is a zero.

The sign convention is crucial here: the factor has a MINUS signStudents see a zero and write the wrong sign in front of the zero.

Let's see this pattern in action:

Zero (α)	Factor (x - α)
$5$	$(x - 5)$
$-3$ tricky!This negative case catches people	$(x - (-3)) = (x + 3)$ You write x minus negative 3, which simplifies
$\frac{1}{2}$	$(x - \frac{1}{2})$

Key observation: When the zero is negativeOpposite signs in the factor, the factor ends up with a plus signThe signs flip when the zero is negative!

✍️ MCQ

Choose one

If

(x + 3)

is a factor of

p(x)

, what is the corresponding zero?

✓

Going backwards (from factor to zero):

Factor (x + 3)The factor tells you the zero → zero is $-3$ Solve by setting the factor to zero (the sign flips)
Factor (x - 5) → zero is $+ 5$

✍️ MCQ

Choose one

If

(x - 7)

is a factor of a polynomial, what is the corresponding zero?

✓

Factor (2x + 7)Same method even with coefficients → set $2x + 7 = 0$ Set factor equal to zero and solve, get $x = - \frac{7}{2}$

✍️ FIB

Fill in the blank

If

(3x - 12)

is a factor of a polynomial, what is the corresponding zero?

✓

The safest approach: always SOLVEThis is when you should reach for this method the equation factor = 0.

Do not try to read off the zero by eyeThis is exactly when sign errors sneak in — that's where sign errorsMake solving your default approach creep in!

Example: If $(x + 3)$ The factor has plus 3, but the zero is negative 3 is a factor, solve:

$x + 3 = 0$

x = -3

zero(This is why solving matters)

So the zero is $-3$ The factor has plus 3, but the zero is negative 3, not $+3$ If you just looked at x plus 3 and guessed, you'd get it wrong.

✍️ MCQ

Choose one

If

(x - 7)

is a factor of a polynomial, what is the corresponding zero?

✓

3. Constant factors do not contribute zeros

When a polynomial is written in factored form like $a (x - α) (x - β)$ , the leading constant $a$ is a factor too.

But constants are never zero, so they do not produce zeros.

Recognising this prevents false zeros — only variable factors like $(x - α)$ contribute zeros.

📋 Given Info

Consider the factored form:

$p (x) = 5 (x - 2) (3 x + 7)$

This has three factors:

$5$ constant
$(x - 2)$
$(3x + 7)$

Not all of them produce zeros.

✍️ Question

Find all zeros of $p(x) = 5(x - 2)(3x + 7)$ .

How many zeros are there, and does the factor $5$ constant contribute one?

In the factored form $p (x) = 5 (x - 2) (3 x + 7)$ , there are three factorsCount the factors — we have three here. Let's examine each one to see which contribute zeros.

Factor 1: 5Constants like 5 can never equal zero

This is a constantThis is a constant factor — it equals 5 no matter what $x$ is. Since 5 is never 0, it does not produce a zeroOnly factors with x can give zeros.

✍️ T/F

True or False?

The constant factor

5

in

p(x) = 5(x-2)(3x+7)

contributes a zero to the polynomial.

✓

Factor 2: $(x - 2)$

Set $x - 2 = 0$ : solving gives $x = 2$ Setting the factor to zero finds the solution. This is a zeroThis is how we find zeros of the polynomial.

Factor 3: $(3x + 7)$

Set $3 x + 7 = 0$ : solving gives $3x = -7$ , so $x = -\frac{7}{3}$ . This is a zero.

✍️ MCQ

Choose one

For the polynomial

p(x) = 5(x - 2)(3x + 7)

, how many zeros are there in total?

✓

Summary: The zeros are $x = 2$ zero 1First of our two zeros and $x = -\frac{7}{3}$ zero 2Second of our two zeros.

Two zeros for a degree-2 polynomialThe degree tells you the maximum number of zeros — exactly as expected!

This connects to the factored form from the previous cluster: $p(x) = a(x - \alpha)(x - \beta)$ The standard way to write any quadratic when you know its zeros.

The ' $a$ stretchIt controls the stretch but can never equal zero' out front ensures the leading coefficient is correct but never produces a zeroIt's just a constant, not a factor with x. Only the $(x - \alpha)$ and $(x - \beta)$ These are the ones that can actually become zero factors do.