Expanding a Term Times a Bracket

How to expand a bracket multiplied by a term with a letter, such as $x(x + 3)$ or $-2a(3a - b)$ , and how to simplify after expanding.

Before you start

In Secondary 1 you expanded a number times a bracket by multiplying each term inside by the number: $3(x + 2) = 3x + 6$ and $-3(6x - y) = -18x + 3y\text{.}$

A minus sign in front of a bracket changes the sign of every term inside: $-(x + y) = -x - y$ and $-(x - y) = -x + y$ .

Expanding means writing a product as a sum or difference of terms. In this lesson the term in front of the bracket has a letter in it.

A Letter in Front of the Bracket

The distributive law

$a(b + c) = ab + ac$ and $a(b - c) = ab - ac$

The term in front multiplies each term inside the bracket. This is true whatever $a$ , $b$ and $c$ stand for: numbers, letters, or terms such as $2x$ .

When $x$ is a length, $x(x + 3)$ is the area of a rectangle $x$ by $(x + 3)$ . Cutting it along the line between $x$ and $3$ gives a square of area $x \times x = x^2$ and a rectangle of area $x \times 3 = 3x$ . The whole area is the sum of the two pieces. This is the shape of Example 9 in the last lesson, where the two areas were added; now we write the whole area as a product and expand it.

Example 1: Expand $x(x + 3)$

Expand $x(x + 3)$ .

Solution

Multiply each term in the bracket by $x$ :

$x(x + 3) = x \times x + x \times 3$

$= x^2 + 3x$

The two terms are the two pieces of the rectangle:

$x(x + 3) = x^2 + 3x$

Multiplying two terms

Multiply the numbers, then the letters. Like signs give $+$ ; unlike signs give $-$ .

$2x \times 3x = 6x^2$ , $\quad -3a \times 4b = -12ab\text{,}$ $\quad -x \times (-5) = 5x$

A letter times itself is a square: $a \times a = a^2$ . And $b \times b^2 = b \times b \times b = b^3\text{.}$

Example 2: Expand $2x(3x - 5)$

Expand $2x(3x - 5)$ .

Solution

$2x(3x - 5) = 2x \times 3x + 2x \times (-5)$

$2x \times 3x$ : $2 \times 3 = 6$ and $x \times x = x^2$ , giving $6x^2$ . $2x \times (-5) = -10x$ .

$= 6x^2 - 10x$

$2x(3x - 5) = 6x^2 - 10x$

A Negative Term in Front

When the term in front is negative, its sign goes with it into every product. Take extra care with the second term: a negative times a negative is positive.

Example 3: Expand $-x(4 - x)$

Expand $-x(4 - x)$ . Write the answer in descending powers of $x$ .

Solution

$-x(4 - x) = (-x) \times 4 + (-x) \times (-x)$

$= -4x + x^2$

In descending powers of $x$ : $x^2 - 4x$ .

$-x(4 - x) = x^2 - 4x$

Example 4: Two negatives, and a minus sign in front

Expand (a) $-2a(-7x - 6y)$ ; (b) $5 - a(-2b + 3c)$ .

Solution

(a) $-2a \times (-7x) = +14ax$ and $-2a \times (-6y) = +12ay\text{.}$

$-2a(-7x - 6y) = 14ax + 12ay$

(b) The $5$ is not multiplied. The term in front of the bracket is $-a$ , sign included:

$-a \times (-2b) = +2ab$ and $-a \times 3c = -3ac$ .

$5 - a(-2b + 3c) = 5 + 2ab - 3ac$

(a) $14ax + 12ay$ ; (b) $5 + 2ab - 3ac$

Powers, Longer Brackets and the Term on the Right

The same law works when the terms have several letters or powers, when the bracket has three terms, and when the term is written after the bracket: $(b + c)a = ba + ca$ .

Example 5: Terms with powers

Expand $ab(ac + b^2)$ .

Solution

$ab \times ac = a \times a \times b \times c = a^2bc$

$ab \times b^2 = a \times b \times b \times b = ab^3$

$ab(ac + b^2) = a^2bc + ab^3$

Example 6: Three terms, and the term on the right

Expand (a) $-2x(x^2 - 3x + 4)$ ; (b) $(3x - 2) \times 4x$ .

Solution

(a) Multiply each of the three terms by $-2x$ :

$-2x \times x^2 = -2x^3$ , $\quad -2x \times (-3x) = +6x^2\text{,}$ $\quad -2x \times 4 = -8x$

$-2x(x^2 - 3x + 4) = -2x^3 + 6x^2 - 8x$

(b) The term $4x$ multiplies each term in the bracket, just as if it were in front:

$(3x - 2) \times 4x = 3x \times 4x - 2 \times 4x = 12x^2 - 8x$

(a) $-2x^3 + 6x^2 - 8x$ ; (b) $12x^2 - 8x$

Expanding and Simplifying

When an expression has two or more products, expand each one, keeping the sign in front of each term outside a bracket with that term. Then collect like terms.

Example 7: Two products with a minus between them

Simplify $-4x(y + 2z) - 3x(5y - z)\text{.}$

Solution

The second term in front of a bracket is $-3x$ , sign included.

$-4x(y + 2z) = -4xy - 8xz$

$-3x(5y - z) = -15xy + 3xz$

Collect: $-4xy - 15xy = -19xy$ and $-8xz + 3xz = -5xz\text{.}$

$-4x(y + 2z) - 3x(5y - z) = -19xy - 5xz$

Example 8: Terms in $x^2$

Simplify (a) $3x(x - 2) - x(2x + 1)\text{;}$ (b) $x(xy + y) - y(xy + x)\text{.}$

Solution

(a) $3x(x - 2) = 3x^2 - 6x$ and $-x(2x + 1) = -2x^2 - x\text{.}$

$3x^2 - 6x - 2x^2 - x = x^2 - 7x$

(b) $x(xy + y) = x^2y + xy$ and $-y(xy + x) = -xy^2 - xy\text{.}$

$x^2y + xy - xy^2 - xy = x^2y - xy^2\text{.}$ The terms $xy$ and $-xy$ cancel, and $x^2y$ and $xy^2$ are unlike, so they stay.

(a) $x^2 - 7x$ ; (b) $x^2y - xy^2$

Working Backwards and Checking

You can check an expansion by putting a number in place of the letter on both sides. If the two values differ, the expansion is wrong. (If they agree, it is very likely right; check every term as well.)

Example 9: The missing term

Find the term that goes in the box: $3x(\square - 4) = 6x^2 - 12x\text{.}$

Solution

The box times $3x$ must give $6x^2$ . Since $3x \times 2x = 6x^2$ , the box is $2x$ .

Check the other term: $3x \times (-4) = -12x$ . ✓

The missing term is $2x$ : $3x(2x - 4) = 6x^2 - 12x\text{.}$

Example 10: Spot the error

Siti wrote (a) $x(x + 3) = x^2 + 3$ and (b) $-2(x - 4) = -2x - 8\text{.}$ Find each mistake and correct it.

Solution

(a) The $3$ was not multiplied by $x$ . Check with $x = 2$ : $2(2 + 3) = 10$ , but $2^2 + 3 = 7$ . Correct: $x(x + 3) = x^2 + 3x$ , and $2^2 + 3(2) = 10$ . ✓

(b) $-2 \times (-4) = +8$ , not $-8$ . Correct: $-2(x - 4) = -2x + 8\text{.}$

(a) $x^2 + 3x$ ; (b) $-2x + 8$

Common Mistakes

Multiplying only the first term in the bracket: $x(x + 3) \ne x^2 + 3$ . Every term is multiplied.
Losing a sign: $-2(x - 4) = -2x + 8\text{,}$ because $-2 \times (-4) = +8$ .
Writing $x \times x$ as $2x$ . It is $x^2$ ; and $2x \times 3x = 6x^2$ , not $6x$ or $5x^2$ .
In $5 - a(-2b + 3c)$ , multiplying by $a$ instead of $-a$ , or multiplying the $5$ as well.
Collecting unlike terms after expanding: $x^2y$ and $xy^2$ are different and cannot be added.

Practice

Try each question before you open its solution. Write answers in descending powers where there is one letter.

Question 1

Expand (a) $4(2x - 3)$ ; (b) $-4(5x - 2y)$ ; (c) $-(5 - 2y)$ .

ShowHide solution

(a) $4 \times 2x - 4 \times 3 = 8x - 12$

(b) $-4 \times 5x = -20x$ and $-4 \times (-2y) = +8y\text{.}$

(a) $8x - 12$ ; (b) $-20x + 8y$ ; (c) $-5 + 2y$ , or $2y - 5$

Question 2

Expand (a) $x(x + 7)$ ; (b) $3y(2y - 1)$ ; (c) $5a(a + 2b)$ .

ShowHide solution

(a) $x \times x + x \times 7 = x^2 + 7x$

(b) $3y \times 2y - 3y \times 1 = 6y^2 - 3y$

(a) $x^2 + 7x$ ; (b) $6y^2 - 3y$ ; (c) $5a^2 + 10ab$

Question 3

Expand (a) $-x(x - 6)$ ; (b) $-4p(3 - 2p)$ ; (c) $-3m(-2m + 5n)$ ; (d) $-2a(3a - b)$ .

ShowHide solution

(a) $-x \times x = -x^2$ and $-x \times (-6) = +6x$ .

(b) $-4p \times 3 = -12p$ and $-4p \times (-2p) = +8p^2\text{.}$ In descending powers: $8p^2 - 12p$ .

(d) $-2a \times 3a = -6a^2$ and $-2a \times (-b) = +2ab\text{.}$

(a) $-x^2 + 6x$ ; (b) $8p^2 - 12p$ ; (c) $6m^2 - 15mn$ ; (d) $-6a^2 + 2ab$

Question 4

Expand (a) $(2x + 5) \times 3x$ ; (b) $3x(x^2 + 2x - 5)$ ; (c) $-a(a^2 - 4a + 1)$ .

ShowHide solution

(a) $2x \times 3x + 5 \times 3x = 6x^2 + 15x$

(b) $3x \times x^2 = 3x^3$ , $3x \times 2x = 6x^2$ , $3x \times (-5) = -15x$ .

(a) $6x^2 + 15x$ ; (b) $3x^3 + 6x^2 - 15x$ ; (c) $-a^3 + 4a^2 - a$

Question 5

Expand (a) $xy(yz + x^2 - xy)$ ; (b) $2pq(p - 3q)$ .

ShowHide solution

(a) $xy \times yz = xy^2z$ , $xy \times x^2 = x^3y$ , $xy \times (-xy) = -x^2y^2\text{.}$

(b) $2pq \times p = 2p^2q$ and $2pq \times (-3q) = -6pq^2\text{.}$

(a) $xy^2z + x^3y - x^2y^2$ ; (b) $2p^2q - 6pq^2$

Question 6

Expand and simplify (a) $7 - 2x(3x - 4)$ ; (b) $x^2 - x(x - 5)$ .

ShowHide solution

(a) The term in front of the bracket is $-2x$ : $-2x \times 3x = -6x^2$ and $-2x \times (-4) = +8x\text{.}$ So $7 - 6x^2 + 8x$ .

(b) $-x \times x = -x^2$ and $-x \times (-5) = +5x$ : $x^2 - x^2 + 5x = 5x$ .

(a) $-6x^2 + 8x + 7$ ; (b) $5x$

Question 7

Simplify (a) $-3x(y + 4z) - 5x(2y - z)\text{;}$ (b) $2p(-4q - 3r) - 6q(3p + 2r)\text{.}$

ShowHide solution

(a) $-3xy - 12xz - 10xy + 5xz\text{.}$ Collect: $-13xy - 7xz$ .

(b) $-8pq - 6pr - 18pq - 12qr\text{.}$ $qp = pq$ , so $-8pq - 18pq = -26pq\text{.}$

(a) $-13xy - 7xz$ ; (b) $-26pq - 6pr - 12qr$

Question 8

Simplify (a) $4x(x - 3) - 2x(x + 5)\text{;}$ (b) $a(a + b) + b(a - b)$ ; (c) $h^2(km + n) - m(h^2m - h^2k)\text{.}$

ShowHide solution

(a) $4x^2 - 12x - 2x^2 - 10x = 2x^2 - 22x$

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

(a) $2x^2 - 22x$ ; (b) $a^2 + 2ab - b^2$ ; (c) $2h^2km + h^2n - h^2m^2$

Question 9

Find the term in each box. (a) $4x(\square + 3) = 8x^2 + 12x\text{;}$ (b) $\square(x - 5) = -2x^2 + 10x\text{;}$ (c) $3a(2a - \square) = 6a^2 - 9ab$

ShowHide solution

(a) $4x \times \square = 8x^2$ , so the box is $2x$ . Check: $4x \times 3 = 12x$ . ✓

(b) $\square \times x = -2x^2$ , so the box is $-2x$ . Check: $-2x \times (-5) = +10x\text{.}$ ✓

(a) $2x$ ; (b) $-2x$ ; (c) $3b$

Question 10

A rectangular garden is $x$ m wide and $(2x + 3)$ m long. (a) Write an expression for its area and expand it. (b) Find the area when $x = 4$ .

ShowHide solution

(a) Area $= x(2x + 3) = (2x^2 + 3x)$ m².

(b) $2(4)^2 + 3(4) = 32 + 12 = 44\text{.}$ Check: the garden is $4$ m by $11$ m, and $4 \times 11 = 44$ . ✓

(a) $(2x^2 + 3x)$ m²; (b) $44$ m²

Question 11

A hall has $r$ rows of seats, with $(2r - 5)$ seats in each row. (a) Write and expand an expression for the number of seats. (b) How many seats are there when $r = 20$ ?

ShowHide solution

(a) Number of seats $= r(2r - 5) = 2r^2 - 5r\text{.}$

(b) $2(20)^2 - 5(20) = 800 - 100 = 700\text{.}$ Check: $20$ rows of $35$ seats is $700$ . ✓

(a) $2r^2 - 5r$ ; (b) $700$ seats

Question 12

Each line has one mistake. Find it and correct it. (a) $3x(x - 4) = 3x^2 - 4$ ; (b) $-2y(y - 5) = -2y^2 - 10y\text{;}$ (c) $x(x + 1) - x(x - 1) = x^2 + x - x^2 - x = 0$

ShowHide solution

(a) The $-4$ was not multiplied by $3x$ : $3x(x - 4) = 3x^2 - 12x\text{.}$

(b) $-2y \times (-5) = +10y\text{:}$ $-2y(y - 5) = -2y^2 + 10y\text{.}$

(c) The term in front of the second bracket is $-x$ , and $-x \times (-1) = +x$ : $x^2 + x - x^2 + x = 2x$ . Check with $x = 3$ : $3(4) - 3(2) = 6 = 2(3)\text{.}$ ✓

(a) $3x^2 - 12x$ ; (b) $-2y^2 + 10y$ ; (c) $2x$

Key Points

$a(b + c) = ab + ac$ : the term in front multiplies every term in the bracket, wherever it is written.
Multiply the numbers, then the letters: $2x \times 3x = 6x^2$ , $a \times a = a^2$ .
The sign of the term in front goes into every product; a negative times a negative is positive.
In $5 - a(\ldots)$ the term multiplying the bracket is $-a$ , and the $5$ is left alone.
After expanding, collect like terms; terms such as $x^2y$ and $xy^2$ are unlike.
Check an expansion by substituting a number on both sides.

Expanding a Term Times a Bracket

Before you start

A Letter in Front of the Bracket

The distributive law

Example 1: Expand x(x+3)x(x + 3)x(x+3)

Multiplying two terms

Example 2: Expand 2x(3x−5)2x(3x - 5)2x(3x−5)

A Negative Term in Front

Example 3: Expand −x(4−x)-x(4 - x)−x(4−x)

Example 4: Two negatives, and a minus sign in front

Powers, Longer Brackets and the Term on the Right

Example 5: Terms with powers

Example 6: Three terms, and the term on the right

Expanding and Simplifying

Example 7: Two products with a minus between them

Example 8: Terms in x2x^2x2

Working Backwards and Checking

Example 9: The missing term

Example 10: Spot the error

Common Mistakes

Practice

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Key Points

Example 1: Expand $x(x + 3)$

Example 2: Expand $2x(3x - 5)$

Example 3: Expand $-x(4 - x)$

Example 8: Terms in $x^2$