Factorise by common factors and difference of perfect squares

Which product is equivalent to the displayed expression?

Combined area

\(\displaystyle 3(2x+5)\)

Move the positive input. Both regions share height 3, so their combined area is the product of the shared height and total width. The model illustrates the distributive identity; multiplying back proves it for every real x.

What becomes the common outside factor?

Undo distribution completely

Factorisation writes a sum as a product without changing its value. Here, fully factorised means using integer coefficients and leaving no shared monomial or applicable difference of perfect squares inside

1. 1Find what every term shares Use the highest common factor of the integer coefficients and the lowest power shared by every term
2. 2Divide every term The terms inside the bracket are the original terms divided by the outside factor
3. 3Keep the requested sign convention A negative outside factor changes every sign inside. A positive outside factor can give an equivalent product, but a labelled table may request one convention
4. 4Check by expanding Multiplying back must reproduce every original term. Then check that no common factor or applicable difference of perfect squares remains

What is the positive highest common factor of the numerical coefficients?

What is the greatest power of x shared by both terms? Enter its exponent

After dividing the first term by 6x², what coefficient remains?

Choose the complete factorisation

Take out −6x. What is the constant term inside the bracket?

Which product uses the requested negative common factor −6x?

Treat the repeated bracket as one object. What is its combined multiplier?

Choose the simplified product

A difference of perfect squares

After removing common factors, look for two squares separated by subtraction. Keep the original subtraction order and use positive square-base coefficients

1. 1Same bases, opposite signs The cross-products cancel when the factors are expanded \(\displaystyle a^2-b^2=(a+b)(a-b)\)
2. 2Check every condition A sum of squares does not use this rule. A squared difference has a middle term and is a different expression

What coefficient belongs to the square base of 9x²?

Choose the complete factorisation

Which product preserves the reversed subtraction order?

Remove the greatest common numerical factor before applying the square identity. What is it?

Choose the fully factorised integer-coefficient form

Can the displayed expression be factorised using a real difference-of-squares identity?