Factorise four-term expressions by grouping

Which grouping makes the same bracket appear after each pair is factorised?

Set the constant c so that both pairs can leave the bracket x + 3. The bars compare values at x = 2. Their difference is c − 6 for every x, so expanding also proves when the two forms agree

Which condition makes the grouping valid?

The same bracket is essential

Grouping is repeated use of the distributive law. A useful grouping leaves identical brackets, not merely brackets with similar-looking terms

1. 1Preserve every term Reorder whole signed terms if needed. Do not change a sign or discard a term
2. 2Factor each pair Divide every term by its pair’s common factor. Choose signs so the remaining brackets match
3. 3Take out the shared bracket Treat the identical binomial as one object. The other factor contains the sum of the outside terms
4. 4Multiply back Check all four products, including any numerical common factor removed at the start

Factor x from the first pair. What constant belongs inside its bracket?

What numerical factor goes outside the second bracket?

Which complete bracket is shared by both terms?

Take out the repeated bracket. Which product results?

Choose the signed common factor that leaves q − 4.

Divide both terms by p. What signed constant remains?

Which product preserves all four signed terms?

The displayed adjacent pairs are unhelpful. Which rearrangement reveals matching brackets?

Choose c so that both pairs leave x + 5.

Which statement about useful grouping is correct?

Extend the same method

The next examples use two variables or a cubic. These extend distributive reasoning. The practice table requests two displayed binomial shapes; a quadratic bracket could sometimes factor further, so this does not mean that no further factorisation is possible

1. 1Keep the displayed shape Factor each pair to leave the same bracket, then take it out \(\displaystyle x^3+2x^2+3x+6=(x^2+3)(x+2)\)
2. 2Keep the requested order and signs Read the monomial in each displayed bracket. Use positive leading coefficients and enter signed constants
3. 3Retain an outside numerical factor If a factor was taken out of all four terms, it still multiplies the final product

What power of x is common to the first two terms? Enter its exponent

Which two-binomial product follows from grouping?

What is the highest positive numerical common factor?

Which equivalent product retains the outside factor?