Welcome! Today we're learning The Quadratic Construction Formula — how to build a polynomial when you already know its zeros.

So far, you've been going from polynomial to zeros (factoring).

Now you reverse the direction: from zeros back to the polynomial.

The construction formula:

p(x) = x^2 - Sx + P

key formula

where:

S = sum of zeros
P = product of zeros

This comes directly from the factored form and is your tool for every 'find the polynomial' problem.

⚠️ Watch out for the double negative!

When S is...	The $- S x$ term becomes...	Common mistake
Positive	Negative coefficient	Usually fine
Negative	Positivetricky! coefficient	Many students get this wrong

1. The construction formula with positive sum

So far, you've been going from polynomial to zeros — that's factoring.

Now we're going to reverse the direction:

Starting from the zeros (or their sum and product) and building the polynomial back up.

Here's the construction formula:

If the sum of zeros is $S$ and the product is $P$ , the quadratic polynomial is:

p(x) = x^2 - Sx + P

KEY

This comes from expanding $(x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha \cdot \beta$

✍️ Question

Your turn! ✏️

Construct a quadratic polynomial whose zeros have:

Sum = 5
Product = 6

Write your answer in the form $p(x) = ...$

The formula to construct a quadratic polynomial from its zeros is:

p(x) = x^2 - Sx + P

(Your go-to tool for finding polynomials from zeros)

where S is the sum of zeros and P is the product of zeros.

Key insight: Notice the negative signcrucial!Minus the sum — where students make mistakes before S and the positive sign+Plus the product — remember this pattern before P. This pattern comes directly from expanding $(x - \alpha)(x - \beta)$ The signs will make sense when you expand this.

✍️ MCQ

Choose one

In the formula

p(x) = x^2 - Sx + P

, what is the sign in front of

S

?

✓

✍️ MCQ

Choose one

Using the formula with

S = 5

and

P = 6

, what is the quadratic polynomial

p(x)

?

✓

With $S = 5$ and $P = 6$ :

p(x) = x^2 - (5)x + 6 = x^2 - 5x + 6

(Take sum and product, plug them directly into this formula)

⚠️ Notice: The formula has a MINUS signThis is where most students lose marks before $S$ . When $S = 5$ , the $x$ -coefficient becomes $-5$ Sum is positive but appears with minus in polynomial.

✍️ MCQ

Choose one

If the sum of zeros is

7

and the product is

10

, what is the coefficient of

x

in the quadratic polynomial?

✓

Verify:Factor it and check if zeros match Factor $x^{2} - 5 x + 6 = (x - 3) (x - 2)$ .

Zeros: 3 and 2.These should match your original zeros

Sum = 3 + 2 = 5Sum should match what you were given ✓

Product = 3 × 2 = 6Product should match original value ✓

Correct!Construction formula confirmed Our polynomial checks out perfectly.

2. The double-negative when S is negative

So far, you've been going from polynomial to zeros (factoring). Now let's reverse direction: from zeros back to the polynomial.

The construction formula is:

p(x) = x^2 - Sx + P

key formula

where:

$S$ = sum of zeros
$P$ = product of zeros

Here's something to watch out for:

When the sum $S$ is negative, the formula produces $- (- something)$ — a double negative.

Example: When $S = - 3$ , the formula gives:

p(x) = x^2 - (-3)x + P

That double negative must be resolved: $-(-3) = +3$ key step

✍️ Question

Your turn ✏️

Construct a quadratic polynomial whose zeros have:

Sum $= - 3$
Product $= - 10$

Write your answer in the form $p (x) = x^{2} + \dots$

When S is negativeWhen S is negative, watch for the trap, the formula creates a double negativeMinus sign before S combined with negative value creates the trap:

$p (x) = x^{2} - S x + P = x^{2} - (- 3) x + (- 10)$

Let's resolve this step by step.

The term $-(-3)$ keyGives you positive 3, not negative 3 needs careful handling — subtracting a negative number is the same as adding the positiveThis is the rule you must remember.

-(-3) = +3

(Not negative 3)

✍️ MCQ

Choose one

After simplifying

-(-3)

, what is the coefficient of

x

in the polynomial?

✓

The $- S$ term: $S = - 3$ , so $-S = -(-3) = +3$ Double negative gives positive — common sign error spot. The coefficient of $x$ is $+ 3$ .

The $+ P$ term: $P = -10$ , so the constant is $-10$ .

Result:

p(x) = x^2 + 3x - 10

final answer(Coefficient of x from negative S, constant from P)

✍️ MCQ

Choose one

In the polynomial

p(x) = x^2 + 3x - 10

, what is the coefficient of

x

?

✓

Verify: Factor $x^2 + 3x - 10 = $(x + 5)(x - 2)$ From this factored form we find our zeros.

Zeros: $-5$ and $2$ .Setting each bracket to zero gives these values

Sum $= -5 + 2 = -3$ ✓Verify the sum matches what we started with

Product $= (-5)(2) = -10$ ✓Verify the product matches too

(verified)Matches!When sum and product match, the polynomial is verified

✍️ MCQ

Choose one

If you construct a quadratic with

S = -4

and

P = -12

, what will be the coefficient of

x

in the polynomial?

✓

Let's see this polynomial we just constructed in action. The graph will show us exactly where those zeros appear — at x = -5 and x = 2 — confirming that our construction formula worked.

✍️ FIB

Fill in the blank

What is the sum of the zeros

-5

and

2

?

✓

💡 The two-step method: Write " $S = - 3$ , so $- S = + 3$ " as a separate line. This makes the sign change visibleWrite it as a separate line so you can see it and helps avoid the double-negative traptrap!Catches many students in exams.

3. Constructing from given zeros (not sum and product)

Sometimes the problem gives the zeros directly instead of the sum and product.

The extra step is:

Compute $S = \alpha + \beta$ (sum of zeros)
Compute $P = \alpha \cdot \beta$ (product of zeros)

Then apply the formula:

p(x) = x^2 - Sx + P

✍️ Question

Construct a quadratic polynomial whose zeros are $3\sqrt{2}$ and $-2\sqrt{2}$ .

When zeros are given directly, the first step is to compute S (sum)Always compute these two values first and P (product)Always compute these two values first.

Our zeros are $3\sqrt{2}$ and $-2\sqrt{2}$ .

We'll use these values to find S and P in the next step.

✍️ MCQ

Choose one

What is

3\sqrt{2} + (-2\sqrt{2})

?

✓

Sum of zeros: $S =$ $<graph elementId="point_zero_1" action="highlight" narrationText="3 root 2">3\sqrt{2}</graph> + <graph elementId="point_zero_2" action="highlight" narrationText="negative 2 root 2">(-2\sqrt{2})</graph>$ (Terms with same surd combine like regular algebra) $=$ $(3 - 2)\sqrt{2}$ Combine like terms just like regular algebra $=$ $\sqrt{2}$ That's our sum

Product of zeros: $P =$ $<graph elementId="point_zero_1" action="highlight" narrationText="3 root 2">(3\sqrt{2})</graph><graph elementId="point_zero_2" action="highlight" narrationText="negative 2 root 2">(-2\sqrt{2})</graph>$ (Multiply the surds together) $= - 6 \times \sqrt{2} \times \sqrt{2} = - 6 \times$ $(\sqrt{2})^2$ That's what square root means $= - 6 \times 2 =$ $-12$ That's our product

✍️ MCQ

Choose one

We found

S = \sqrt{2}

and

P = -12

. Using the construction formula

p(x) = x^2 - Sx + P

, what will be the coefficient of

x

in our polynomial?

✓

Now we construct the polynomial using $p(x) = x^2 - Sx + P$ S is the sum, P is the product, in these positions:

$p (x) = x^{2} - (\sqrt{2}) x + (- 12) = x^{2} - \sqrt{2} x - 12$

Notice the coefficient of $x$ is $-\sqrt{2}$ irrationalWhen zeros have surds, the polynomial will have surds too, which is irrational. That is perfectly fine — not every constructed polynomial has integer coefficientsYour coefficients don't have to be integers.

✍️ MCQ

Choose one

If the zeros of a quadratic polynomial are

5\sqrt{3}

and

-\sqrt{3}

, what is the sum of zeros

S

?

✓

Let's see how our constructed polynomial actually passes through the zeros we started with. The graph confirms that the formula works perfectly.

✍️ FIB

Fill in the blank

The parabola opens ______, which means the coefficient of

x^2

is ______.

✓

Verification: To confirm our construction is correct, let's factor $x^{2} - \sqrt{2} x - 12$ .

We need two numbers with product $= -12$ These conditions come from Vieta's formulas and sum $= \sqrt{2}$ The sum and product determine your zeros.

The numbers $3\sqrt{2}$ and $-2\sqrt{2}$ The zeros match what we started with work:

Product: $(3 \sqrt{2}) (- 2 \sqrt{2}) = - 12$ ✓
Sum: $3 \sqrt{2} + (- 2 \sqrt{2}) = \sqrt{2}$ ✓

🎯 Wait — these are exactly our original zeros!Matching zeros confirms your work This confirms our construction is correct — $x^{2} - \sqrt{2} x - 12$ .

✍️ MCQ

Choose one

If you construct a quadratic polynomial using

S = 4

and

P = -5

, and then factor it back, what should the zeros multiply to give?

✓