Learn: Polynomials & Algebraic Identities

Review the key ideas, then practise whenever you’re ready.

The idea

A polynomial is an expression built from powers of $x$ , like $3x^2 + 5x - 2$ . Think of brackets as unopened gift boxes: expanding unpacks the brackets into separate terms, and factoring packs them back into brackets. Three area identities let you expand and factor quadratics in seconds.

The four building blocks:

1
Combine like terms only - only add terms with the exact same power of $x$ ( $3x^2 + 5x^2 = 8x^2$ , while $x^2$ and $x$ stay separate).
2
FOIL two brackets - multiply all 4 pairs: First ( $x \cdot x$ ) + Outer ( $x \cdot b$ ) + Inner ( $a \cdot x$ ) + Last ( $a \cdot b$ ).
3
The 3 Golden Identities - $(a+b)^2 = a^2 + 2ab + b^2$ , $(a-b)^2 = a^2 - 2ab + b^2$ , and $(a-b)(a+b) = a^2 - b^2$ .
4
Remainder & Factor Shortcut - dividing $P(x)$ by $(x - r)$ leaves remainder $P(r)$ . If $P(r) = 0$ , then $(x - r)$ is an exact factor!

Quick memory tips

Squaring a bracket ALWAYS gives 3 terms - never forget the middle $2ab$ twin rectangles
Mental-math trick: $53^2 - 47^2 = (53 - 47)(53 + 47) = 6 \times 100 = 600$
Remainder Theorem sign flip: dividing by $(x - 2)$ means plugging in $x = 2$

Interactive Area & Identity Explorer

Click each identity to see the 4 pieces of the multiplication grid.

4-box multiplication grid

Why (a + b)² Has a Middle +2ab Term

Squaring (x + 3) creates four boxes, not two! The two amber corner strips (+3x and +3x) are identical, so they team up to make +6x.

(x + 3)^2 = x^2 + 6x + 9

The 3 Golden Identities at a Glance

1. Square of a Sum

(a+b)^2 = a^2 + 2ab + b^2

2. Square of a Difference

(a-b)^2 = a^2 - 2ab + b^2

3. Difference of Two Squares

(a-b)(a+b) = a^2 - b^2

Worked example

Every step explained - why we do it, and what happens.

1
$(2x - 3)^2$
Why: Use the Square of a Difference identity (A − B)² = A² − 2AB + B² with first piece A = 2x and second piece B = 3.
What happens: Three slots to fill: first squared, minus twice the product, plus last squared.
2
$= (2x)^2 - 2(2x)(3) + 3^2$
Why: Write out all three parts clearly before simplifying. Notice the last sign is + because (−3) × (−3) = +9.
What happens: Every piece of the identity is in place.
3
$(2x)^2 = 4x^2$
Why: When squaring (2x), square BOTH the number 2 and the letter x: 2² · x² = 4x².
What happens: The leading term is 4x² (not 2x²).
4
$= 4x^2 - 12x + 9$
Why: Multiply the middle numbers (−2 × 2x × 3 = −12x) and square the last number (3² = 9).
What happens: Fully expanded: 4x² − 12x + 9.
5
$4(5)^2 - 12(5) + 9 = 100 - 60 + 9 = 49$
Why: The check: test x = 5. The original bracket gives (2(5) − 3)² = 7² = 49, and our expanded expression also gives 49.
What happens: 49 = 49 ✓ - the expansion is exact.

Remainder & Factor Theorem (Optional Toolbox)

How to find the remainder of a long division in 10 seconds - without writing a single division bracket!

Levels 4–6

Why the shortcut works

Any division can be written as Polynomial = (Divisor × Quotient) + Remainder:

P(x) = (x - r)\cdot Q(x) + R

Look what happens when you plug in x = r: the bracket $(r - r)$ becomes 0, which wipes out $Q(x)$ completely and leaves P(r) = R!

1. Remainder Theorem

To find the remainder when $P(x)$ is divided by $(x - r)$ , flip the sign in the bracket ( $x - r = 0 \Rightarrow x = r$ ) and compute $P(r)$ .

2. Factor Theorem

If the remainder is zero ( $P(r) = 0$ ), then $(x - r)$ divides evenly with nothing left over - it is an exact factor!

Real-world example

A square garden has side length 20 m (area = 400 m²). You lengthen the north-south sides by 5 m (to 25 m) and shorten the east-west sides by 5 m (to 15 m). Does the area stay 400 m², or does it change?

1
$\text{Area} = (x - 5)(x + 5)$
Why: With original side x = 20, the new rectangle has sides (x − 5) and (x + 5).
What happens: This is the Difference of Two Squares pattern: (a − b)(a + b).
2
$= x^2 + 5x - 5x - 5^2 = x^2 - 25$
Why: The outer (+5x) and inner (−5x) rectangles cancel each other out completely.
What happens: The new area is always 25 m² LESS than the original x² square!
3
$20^2 - 25 = 400 - 25 = 375 \text{ m}^2$
Why: Substitute x = 20 into x² − 25.
What happens: The new rectangular garden has area 375 m².
4
$\text{Trap: } +5 \text{ m and } -5 \text{ m cancel} \Rightarrow 400 \text{ m}^2\,?$
Why: The glitch: thinking that adding 5 m one way and taking 5 m the other way keeps the area at 400 m². Only the perimeter stays the same - the area loses 5² = 25 m²!
What happens: Area is 375 m², not 400 m².
5
$15 \times 25 = 375 \text{ m}^2$
Why: The check: multiply the actual new sides 15 m × 25 m directly.
What happens: 15 × 25 = 375 m² ✓ - the identity gave the exact answer in one step.

Common glitch

Three classic polynomial glitches:

1.Dropping the middle $2ab$ term when squaring a bracket

Wrong: $(x + 3)^2 = x^2 + 9$ - missing the two $3x$ corner rectangles in the area grid

Right:Squaring a binomial always produces 3 terms: $(x + 3)^2 = x^2 + 6x + 9$ .

2.Making the last term negative in $(a - b)^2$

Wrong: $(x - 3)^2 = x^2 - 6x - 9$ - forgetting that $(-3) \times (-3) = +9$

Right:A negative squared is always positive: $(x - 3)^2 = x^2 - 6x + 9$ .

3.Plugging in the wrong sign in the Remainder Theorem

Wrong:Dividing by $(x - 3)$ and plugging in $x = -3$

Right:Set the divisor to zero first ( $x - 3 = 0 \Rightarrow x = 3$ ) and evaluate $P(3)$ .

Quadratic Equation Solver - Check quadratic expansions, factoring, and roots step by step.
Linear & Quadratic Graphing Calculator - Plot a polynomial to see how its coefficients shape the curve.

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Common glitch

Three classic polynomial glitches:

1.Dropping the middle $2ab$ term when squaring a bracket

Wrong: $(x + 3)^2 = x^2 + 9$ - missing the two $3x$ corner rectangles in the area grid

Right:Squaring a binomial always produces 3 terms: $(x + 3)^2 = x^2 + 6x + 9$ .

2.Making the last term negative in $(a - b)^2$

Wrong: $(x - 3)^2 = x^2 - 6x - 9$ - forgetting that $(-3) \times (-3) = +9$

Right:A negative squared is always positive: $(x - 3)^2 = x^2 - 6x + 9$ .

3.Plugging in the wrong sign in the Remainder Theorem

Wrong:Dividing by $(x - 3)$ and plugging in $x = -3$

Right:Set the divisor to zero first ( $x - 3 = 0 \Rightarrow x = 3$ ) and evaluate $P(3)$ .

Learn: Polynomials & Algebraic Identities

The idea

The four building blocks:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Polynomials & Algebraic Identities

The idea

The four building blocks:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator