FA4, HA4 - Expanding two brackets

FA4, HA4 - Expanding two brackets

Learn to multiply two brackets, explain why a middle term sometimes disappears, and check an expansion before trusting it. This develops the two-binomial expansion skills in FA4 and HA4 and prepares you to reverse the process when factorising.

Two binomials

Does (x+3)(x+2)(x+3)(x+2) equal x2+6x^2+6? Multiplying the first terms and the last terms misses two products. A rectangle helps us see where they come from.

For positive xx, a rectangle with sides x+3x+3 and x+2x+2 splits into four smaller rectangles. Their areas are x2x^2, 3x3x, 2x2x and 66. Adding all four gives the whole area.

[DIAGRAM: asset_name: Four products in two brackets; asset_slug: four-products-area; caption: Every part of one side multiplies every part of the other side.; description: An exact partitioned rectangle has horizontal side sections x and 3 and vertical sections x and 2. Its four regions are labelled x squared, 3x, 2x and 6, and the total area is x squared plus 5x plus 6. The area model assumes x is positive.]
Every part of one side multiplies every part of the other side.

Binomial

A binomial has two terms joined by addition or subtraction, such as x+3x+3 or 2x−52x-5. To expand a product of two binomials, multiply each term in the first bracket by each term in the second, then collect like terms.

The same rule follows from distributing twice:

(a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd(a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd

Although an ordinary area picture uses positive lengths, the distributive law works with negative terms too. Keep each sign attached to its term.

Worked example 1: four positive products

Question. Expand and simplify (x+3)(x+2)(x+3)(x+2).

Plan. Multiply xx and 33 by both terms of the second bracket.

(x+3)(x+2)=x2+2x+3x+6four term pairs=x2+5x+6collect the x terms\begin{aligned} (x+3)(x+2)&=x^2+2x+3x+6 &&\text{four term pairs}\\ &=x^2+5x+6 &&\text{collect the }x\text{ terms} \end{aligned}

Answer. x2+5x+6x^2+5x+6.

Check. At x=2x=2, the brackets give 5×4=205\times4=20. The expansion gives 4+10+6=204+10+6=20.

Now change only the sign before 22. Two of the four products change sign.

Worked example 2: a negative term

Question. Expand and simplify (x+3)(x−2)(x+3)(x-2).

Plan. Treat the second bracket as x+(−2)x+(-2) and form all four products.

(x+3)(x−2)=x2−2x+3x−6multiply with signs=x2+x−6combine −2x+3x\begin{aligned} (x+3)(x-2)&=x^2-2x+3x-6 &&\text{multiply with signs}\\ &=x^2+x-6 &&\text{combine }-2x+3x \end{aligned}

Answer. x2+x−6x^2+x-6.

Check. At x=1x=1, the brackets give 4×(−1)=−44\times(-1)=-4. The expansion gives 1+1−6=−41+1-6=-4.

The middle terms count different products even though both contain xx.

Watch out

(x+3)(x+2)=x2+6(x+3)(x+2)=x^2+6 omits 2x2x and 3x3x. Write all four products before collecting; the correct expansion is x2+5x+6x^2+5x+6.

Keep the sign change in mind as you try a fresh pair.

Difference of two squares

When two brackets have the same terms but opposite signs between them, their middle products cancel.

Start with (x+5)(x−5)(x+5)(x-5). The middle products are −5x-5x and +5x+5x, which add to zero. This leaves x2−25x^2-25, a subtraction of two square terms.

Difference of two squares

(a+b)(a−b)=a2−b2(a+b)(a-b)=a^2-b^2

The two brackets must have the same first term and the same second term, with opposite signs between them. This works for any real aa and bb.

Not in the formula booklet: learn it.

You can always expand all four products if you are unsure whether this structure applies.

Worked example 3: cancelling middle terms

Question. Expand and simplify (x+5)(x−5)(x+5)(x-5).

Plan. Show the two equal and opposite middle products.

(x+5)(x−5)=x2−5x+5x−25distribute=x2−25middle terms cancel\begin{aligned} (x+5)(x-5)&=x^2-5x+5x-25 &&\text{distribute}\\ &=x^2-25 &&\text{middle terms cancel} \end{aligned}

Answer. x2−25x^2-25.

Check. At x=7x=7, the product is 12×2=2412\times2=24, and 49−25=2449-25=24.

Next replace the first term xx by 3x3x. The same cancellation occurs, but the whole first term must be squared.

Continue this lesson.

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