Expand and factorise linear expressions

Which expression is equivalent to the supplied bracket for every x?

Distribute a number to each signed term

Multiply every term inside the brackets by the outside numerical factor. Keep each sign with its term

The numerical factors here do not contain variables. Multiplying a linear term by a number changes its coefficient, not its variable power

Enter signed integer coefficients in the requested order. A constant has no variable, and an unprinted coefficient before a letter is one

Multiply both terms by a positive number

Expand the bracket and enter the coefficients of the labelled terms

Enter signed integers in this order: coefficient of x, constant term. Enter 0 for any group that cancels completely

Multiply every bracket term by the signed outside factor

Evaluate the signed products

Expand a difference using a positive factor

Include the sign of the outside factor

A negative outside factor multiplies every signed term. Negative times positive is negative, and negative times negative is positive

A minus sign before brackets means a factor of negative one, not an instruction to change only the first term

Multiply a positive sum by a negative factor

Expand the bracket and enter the coefficients of the labelled terms

Enter signed integers in this order: coefficient of x, constant term. Enter 0 for any group that cancels completely

Multiply every bracket term by the signed outside factor

Evaluate the signed products

Keep both negative multiplication signs

Expand a bracket with factor negative one

Retain two different variable groups

Expand the bracket and enter the coefficients of the labelled terms

Enter signed integers in this order: coefficient of a, coefficient of b. Enter 0 for any group that cancels completely

Multiply every bracket term by the signed outside factor

Evaluate the signed products

Distribute without combining unlike variables

Keep an outside addend separate from a multiplier

The leading seven is an addend. It is not part of the negative two multiplying the bracket

Expand first, then collect constants together and coefficients of the same variable together

A requested group may cancel to zero. Enter zero in its table box rather than leaving it empty

Expand before combining the constant terms

Expand and collect like terms, then enter the coefficients

Enter signed integers in this order: coefficient of x, constant term. Enter 0 for any group that cancels completely

Keep the leading constant and distribute the signed outside factor

Evaluate the signed products

Group the coefficients of like terms, including the constants

Add the signed coefficients in each group; omit any zero terms

Retain a negative leading constant

Record a constant group that cancels

Expand each numerical multiple before collecting

Treat the second outside factor as negative three, and distribute it to both terms in its own bracket

Then collect the x coefficients separately from the constants

The two brackets are added or subtracted here, not multiplied by each other. These expressions remain linear

If all groups cancel, the whole expression is zero; otherwise retain every uncancelled group

Expand a sum of two numerical multiples

Expand and collect like terms, then enter the coefficients

Enter signed integers in this order: coefficient of x, constant term. Enter 0 for any group that cancels completely

Distribute the first outside factor; keep the second bracket unchanged

Evaluate the signed products

Multiply every bracket term by the signed outside factor

Evaluate the signed products

Group the coefficients of like terms, including the constants

Add the signed coefficients in each group; omit any zero terms

Collect after two signed expansions

Keep a constant when the variable terms cancel

Account for cancellation in every requested group

Distinguish checking from justifying an identity

Distributivity justifies this equality for every k and x. Here k is the chosen numerical outside factor

Substitution is useful for detecting an error. One mismatch disproves an identity, but several matching examples alone do not prove it for every input

Reversing the distributive equality writes a sum as a common factor times a bracket. This is factorisation

Product

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

Expanded form

\(\displaystyle 2kx+3k\)

Choose integer values for the numerical factor k and the variable x. The table compares the product and expanded forms, then subtracts their values

Why do the two forms agree beyond the displayed examples?

Reverse distribution using a common numerical factor

The positive highest common factor of six and fifteen is three. Divide each original term by three to recover both bracket terms

Unless a negative outside factor is explicitly requested, use the positive HCF of the coefficient magnitudes

The outside factor multiplies every term in the bracket. Multiply back to verify that every original sign and coefficient is recovered

Coefficient magnitudes

\(\displaystyle 12,\ 18\)

Enter the positive HCF of the two coefficient magnitudes

List common positive factors or compare prime factorizations

Construct the outside factor and every bracket coefficient

The factorisation tables ask for the outside numerical factor first, followed by each bracket coefficient in the named order

Here the three entries are six, two and negative three. Dividing the constant term must retain its negative sign

Do not enter the whole expression in a coefficient box. The variable letters and bracket structure are supplied; your signed integers complete them

Extract the positive HCF from a sum

Use the linear expression below

Factorise the expression using its numerical common factor

Use the positive HCF as the outside factor

Enter signed integers in this order: outside factor, coefficient of x inside the brackets, constant term inside the brackets

The complete factorisation is:

The products restore the original expression:

Construct a positive outside factor and bracket

Retain a negative constant inside the bracket

Use the linear expression below

Factorise the expression using its numerical common factor

Use the positive HCF as the outside factor

Enter signed integers in this order: outside factor, coefficient of x inside the brackets, constant term inside the brackets

The complete factorisation is:

The products restore the original expression:

Factorise a difference by its positive HCF

Retain a negative variable coefficient

A positive outside factor can leave two negative terms

Use the linear expression below

Factorise the expression using its numerical common factor

Use the positive HCF as the outside factor

Enter signed integers in this order: outside factor, coefficient of x inside the brackets, constant term inside the brackets

The complete factorisation is:

The products restore the original expression:

Choose a negative outside factor when requested

Both products are equivalent to the original expression. The requested outside sign decides which form belongs in the answer table

When the negative of the positive HCF is requested, divide every original term by that negative number

Do not change only the outside sign while keeping the bracket unchanged; that would negate the entire expression

Extract a negative common factor from two negative terms

Use the linear expression below

Factorise the expression using its numerical common factor

Use the negative of the positive HCF as the outside factor

Enter signed integers in this order: outside factor, coefficient of x inside the brackets, constant term inside the brackets

The complete factorisation is:

The products restore the original expression:

Use the explicitly requested negative outside factor

Different variables can share a numerical factor

The common numerical factor is three. The x and y remain with their original terms

Different variables do not become a common variable factor merely because both terms contain a letter

This task removes only the greatest common numerical factor from two linear terms; it does not multiply the variables together

Extract a numerical factor while retaining unlike terms

Use the linear expression below

Factorise the expression using its numerical common factor

Use the positive HCF as the outside factor

Enter signed integers in this order: outside factor, coefficient of x inside the brackets, coefficient of y inside the brackets

The complete factorisation is:

The products restore the original expression:

Construct two signed variable coefficients inside the bracket

Recognise when the positive HCF is one

If the coefficient magnitudes have no common positive factor greater than one, they are coprime

A table requesting the positive HCF still has a complete answer: enter one outside and retain both original bracket coefficients

In ordinary written algebra, multiplying by one is usually omitted. There is no non-trivial numerical factor to remove in this case

Record an identity factor without changing either term

Original expression

\(\displaystyle 6x+18\)

Which form uses the positive HCF of the two original coefficients as its outside factor?

Identity to complete

\(\displaystyle 12x-18=6(2x+c)\)

Find the constant c that makes this identity true for every x

Divide the original constant term by the outside factor

Expression

\(\displaystyle -6(2x+3)\)

Input

\(\displaystyle x=2\)

Evaluate the factorised expression at the given value of x

Evaluate the bracket first, then multiply its whole value by negative six

Create a reversible pair of linear expressions

Independent written task, not automatically marked

Choose two non-zero coprime integers and use one as a variable coefficient and the other as a constant inside a bracket. Choose an outside integer with magnitude greater than one

Expand your product, then recover its greatest common numerical factor from the resulting coefficients. State whether the requested outside factor is positive or negative

Explain why every term must be multiplied during expansion and divided during factorisation. Check your pair at a negative input and justify it using distributivity

Create a second example with two unlike variables, and retain both variables through the reverse steps

Review the construction and explanations against these instructions or with a teacher; the coefficient tables check the supplied expressions rather than your independent creation

Reverse the form without changing the expression

Distribute a signed numerical multiplier to every bracket term, then collect only matching variable groups and constants

Keep leading addends separate from outside multipliers. Enter zero for requested groups that cancel

To factorise, find the HCF of the coefficient magnitudes, choose the requested outside sign and divide every term by that signed factor

The positive HCF form and a specifically requested negative-factor form can both represent the same expression

Multiply back to verify every coefficient and sign. A numerical substitution checks an example; distributivity establishes the identity

When creating your own example, state the original terms, the chosen factor and the reason each reverse step preserves the value