Welcome! Today we're looking at Verifying Division Results and Finding What to Subtract — two essential skills that will save you from careless errors.

After every division, you should verify the result by checking:

$q \times g + r = f$

This expansion catches errors and is itself a scored step in exams.

Beyond verification, the division algorithm leads to a practical question:

Question	Answer
What do you subtract from the dividend to make the division exact?	The remainder

1. Verifying a division result by expansion

Verification Time ✅

After performing polynomial division, there's an important step that both catches errors and earns you marks: verification.

Verification means expanding $q \times g + r$ and checking that it equals the original dividend $f$ .

This confirms your division was done correctly.

📋 Given Info

Here's what we have:

Division of $5 x^{3} - 13 x^{2} + 21 x - 14$ by $(x^{2} - 2 x + 3)$ gives:

Component	Value
Quotient	$(5x - 3)$
Remainder	$-5$

Let's see if you can verify this result.

✍️ Question

Verify the division result 🔍

Verify:

(5x - 3)(x^2 - 2x + 3) + (-5) = 5x^3 - 13x^2 + 21x - 14

Show the full expansion.

Let's expand $(5 x - 3) (x^{2} - 2 x + 3)$ term by term.

We'll multiply each term of $(5x - 3)$ First bracket terms multiply with all second bracket terms with each term of $(x^2 - 2x + 3)$ Don't miss any combination.

First, multiply $5x$ with each term of $(x^{2} - 2 x + 3)$ :

$5x \times x^2 = 5x^3$

$5x \times (-2x) = -10x^2$

$5x \times 3 = 15x$

✍️ MCQ

Choose one

When we multiply

(-3)

by

(-2x)

, what do we get?

✓

Next, multiply $(- 3)$ with each term of $(x^{2} - 2 x + 3)$ :

$(- 3) \times x^{2} = - 3 x^{2}$

$(-3) \times (-2x) = 6x$ signs!Result is positive — most sign errors happen here

$(- 3) \times 3 = - 9$

Now combine like termsGrouping terms with the same power of x together:

$5 x^{3} + (- 10 x^{2} - 3 x^{2}) + (15 x + 6 x) + (- 9)$

$5x^3$ alone(No other x cubed term to combine it with) $(-10x^2 - 3x^2)$ All x squared terms grouped together $(15x + 6x)$ All x terms in another group $(- 9)$

✍️ MCQ

Choose one

What is

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

?

✓

$= 5x^3 - 13x^2 + 21x - 9$

$-13x^2$ $+21x$ $-9$ const

Add the remainderAlways add the remainder, even when negative $(- 5)$ :

$5 x^{3} - 13 x^{2} + 21 x - 9 + (- 5) = 5 x^{3} - 13 x^{2} + 21 x - 9 - 5$

✍️ MCQ

Choose one

What is

-9 + (-5)

?

✓

= 5x^3 - 13x^2 + 21x - 14

(This confirms your division is correct)

This equals the original dividendMust match exactly to verify. Verified.Use this check on tests ✓

2. What to subtract for exact divisibility

Let's explore an interesting application of the division algorithm! 🔍

You know that when we divide a polynomial $f (x)$ by $g (x)$ , we get:

f(x) = q(x) \cdot g(x) + r(x)

Now here's a practical question: What if we want the division to be exact? That is, what if we want the remainder to be zero?

If we subtract $r (x)$ from both sides of the equation:

f(x) - r(x) = q(x) \cdot g(x)

This new expression is exactly divisible by $g (x)$ — no remainderkey result!

✍️ Question

Problem 📝

When $3 x^{3} + 10 x^{2} - 14 x + 9$ is divided by $(3 x - 2)$ , the remainder is $5$ .

What must be subtracted from the polynomial so that $(3 x - 2)$ divides it exactly?

Explain the logic behind your answer.

The division algorithmYour go-to formula for checking division says:

f(x) = q(x) \cdot g(x) + R

(This relationship must hold when dividing polynomials)

where $R$ is the remainder (here $R = 5$ ).

✍️ MCQ

Choose one

In the equation

f(x) = q(x) \cdot g(x) + R

, if we want

g(x)

to divide

f(x)

exactly, what should the remainder

R

be?

✓

If you subtract $R$ from $f (x)$ :

f(x) - R = q(x) \cdot g(x) + R - R = q(x) \cdot g(x)

(Subtracting R cancels the remainder term)

The right side is a multiple of $g(x)$ What remains after subtracting R — meaning $g(x)$ divides it exactlyClean division with no leftover (remainder = 0That's what divides perfectly means).

✍️ MCQ

Choose one

What must be subtracted from

3x^3 + 10x^2 - 14x + 9

so that

(3x - 2)

divides it exactly?

✓

So the answer to 'what must be subtracted' is always: subtract the remainderThe remainder from division is exactly what you subtract.

Here: subtract 5answer.

The new polynomial $(3x^3 + 10x^2 - 14x + 9 - 5) = (3x^3 + 10x^2 - 14x + 4)$ is exactly divisible by $(3 x - 2)$ .

✍️ MCQ

Choose one

If dividing a polynomial by

(x + 3)

gives remainder

-7

, what must be subtracted to make it exactly divisible?

✓

3. Finding g(x) from the division algorithm equation

Sometimes the problem gives you $f (x)$ , $q (x)$ , and $r (x)$ and asks you to find the divisor $g(x)$ .

Rearranging the division algorithm $f = q \cdot g + r$ gives us:

g(x) = \frac{f(x) - r(x)}{q(x)}

Approach: First subtract $r$ from $f$ , then divide the result by $q$ .

✍️ Question

Find $g (x)$ 📝

Given:

$f(x) = x^3 - 3x^2 + x + 2$
$q(x) = x - 2$
$r(x) = -2x + 4$

Find $g(x)$ find this. Show the key steps.

The procedure for finding $g (x)$ :

Step 1: Compute $f(x) - r(x)$ Removes the leftover part for perfect division.

f(x) - r(x) = (x^3 - 3x^2 + x + 2) - (-2x + 4)

(You're subtracting a whole expression, not just individual terms)

(subtracting the entire remainder)This is where most students make mistakes

✍️ MCQ

Choose one

When we subtract

(-2x + 4)

from a polynomial, what happens to the

-2x

term?

✓

$= x^{3} - 3 x^{2} + x + 2 + 2 x - 4$

(Notice: subtracting $-2x$ gives $+2x$ Sign changes are critical here, and subtracting $+4$ gives $-4$ Get this wrong and your answer falls apart)

= x^3 - 3x^2 + 3x - 2

result(Once we removed the remainder, this divides evenly)

This is $f (x) - r (x)$ , which we will divide by $q(x)$ No remainder left after this division in the next step.

Step 2: Divide by $q (x) = (x - 2)$ .

This is a standard long divisionSame process you've used before:

$x^3 \div x = x^2$ How you find each term of the quotient. Multiply: $x^{2} (x - 2) = x^{3} - 2 x^{2}$ . Subtract: $-x^2 + 3x - 2$ resultThis becomes your new dividend for the next round.

$-x^2 \div x = -x$ Always divide the leading terms to get the next quotient term. Multiply: $- x (x - 2) = - x^{2} + 2 x$ . Subtract: $x - 2$ result(Each step reduces the degree by one).

✍️ MCQ

Choose one

We have

x - 2

remaining and we're dividing by

x - 2

. What will be the next term in the quotient?

✓

$x \div x = 1$ Simple as that. Multiply: $1 (x - 2) = x - 2$ . Subtract: $0$ Confirms our earlier subtraction was correct.

Remainder is $0$ Proves our setup worked (confirming the setup is correctIf we got a non-zero remainder, something would have gone wrong).

g(x) = x^2 - x + 1

answer(That's the polynomial we were looking for)

✍️ T/F

True or False?

To verify: if

g(x) = x^2 - x + 1

,

q(x) = x - 2

, and

r(x) = -2x + 4

, then

q(x) \cdot g(x) + r(x)

should equal

f(x) = x^3 - 3x^2 + x + 2

. True or False?

✓

Key Rule: Subtract $r(x)$ BEFORE dividing.The correct order: subtract first, divide second

If you forget to subtract the remainder first, your division will have a non-zero remainder — and you'll get the wrong answerReversing the order gives incorrect results for $g (x)$ .

The order matters:

First: $f(x) - r(x)$ Step 1(Always do subtraction before division)
Then: Divide by $q(x)$ Step 2Division comes after subtraction

Never skip step 1!Skipping subtraction costs marks in exams

✍️ MCQ

Choose one

A student tries to find

g(x)

by directly dividing

f(x)

by

q(x)

without subtracting

r(x)

first. What will happen?

✓