Expanding binomials and taking out common factors – Year 9 maths worksheet with answers to download as PDF

Multiply every term inside the bracket by the term outside. Expand each expression.

1. \(3(x+5)\)
2. \(4(2y-3)\)
3. \(-2(a+6)\)
4. \(x(x+7)\)

Expand the brackets, then collect like terms. Show the expanded line before you simplify.

1. \(3(x+2)+2(x+5)\)
2. \(5(2a-1)-3(a+4)\)
3. \(4(y+3)-(y-2)\)
4. \(x(x+3)+2(x+1)\)

Expand and simplify. Use the grid method or the four-term method (first, outside, inside, last).

1. \((x+3)(x+5)\)
2. \((x+4)(x-2)\)
3. \((x-3)(x-6)\)
4. \((2x+1)(x+4)\)

Remember that \((x+a)^2\) means \((x+a)(x+a)\), not \(x^2+a^2\).

1. \((x+4)^2\)
2. \((x-5)^2\)
3. \((x+6)(x-6)\)
4. \((3x+2)^2\)

Factorise each expression fully: take the highest common factor outside a bracket.

1. \(6x+9\)
2. \(8y-12\)
3. \(x^2+5x\)
4. \(12ab+18a\)

Expanding is the reverse of factorising, so you can check a factorised answer. Answer these questions.

1. Is \(5(x+2)=5x+2\)? Expand to check.
2. Factorise \(2x^2+6x\) and check by expanding.
3. Factorise \(x^2-49\).
4. Factorise \(4x^2-9\) using the difference of two squares.

The school garden is a rectangle measuring \((x+6)\) metres by \((x+2)\) metres. Mrs Evans wants to know its area and the cost of fencing.

1. Show that the area is \(x^2+8x+12\) square metres.
2. Find the area when \(x=4\) and check using the original lengths.
3. Write a simplified expression for the perimeter.
4. Fencing costs £8.50 per metre. Find the cost when \(x=4\).

The school hall is \((2x+1)\) metres long and \((x+4)\) metres wide. The caretaker is ordering floor varnish.

1. Show that the area of the floor is \(2x^2+9x+4\) square metres.
2. Find the area when \(x=5\) using the expression and using the lengths.
3. One tin of varnish covers 20 m². How many tins are needed when \(x=5\)?
4. Each tin costs £14.20. Find the total cost.

Ali has a square picture with sides of \(x\) cm. He puts it in a square frame so that there is a 3 cm border on every side.

1. Write an expression for the side length of the whole frame.
2. Expand the area of the whole frame.
3. Show that the area of the border alone is \(12x+36\).
4. Find the area of the border when \(x=20\) and check with the numbers.

At the school fair, a child ticket costs £\(x\) and an adult ticket costs £\((x+2)\). The Brown family buys 2 adult tickets and 3 child tickets.

1. Write and simplify an expression for the total cost.
2. Find the total cost when \(x=3\).
3. The Green family buys 4 adult tickets and 6 child tickets. Write and simplify the total cost, then factorise it.
4. What does your factorised answer tell you about the two families?

Leila's teacher shows the class how to use algebra for quick calculations, such as \(103\times97=(100+3)(100-3)\).

1. Use the difference of two squares to work out \(103\times97\).
2. Work out \(52^2\) by writing it as \((50+2)^2\).
3. Work out \(48\times52\).
4. Work out \(99^2\).

A baker packs each box with \(x\) muffins and 3 cookies. Each day she fills 4 boxes in the morning and 6 boxes in the afternoon.

1. Write an expression for the number of items in one box.
2. Write and simplify an expression for the total number of items packed in a day.
3. Factorise your answer.
4. Find the total number of items when \(x=9\).

A factory makes a cuboid box with dimensions \(x\) cm, \((x+2)\) cm and \((x+3)\) cm.

1. Expand \((x+2)(x+3)\).
2. Hence show that the volume is \(x^3+5x^2+6x\) cm\(^3\).
3. Find the volume when \(x=4\) both ways.
4. The box is made of card costing 0.2p per cm\(^2\) of surface area. When \(x=4\) the surface area is 188 cm\(^2\). Find the cost of the card.

Jade saves £\((2x+5)\) each week for 4 weeks. Ben saves £\((x+8)\) each week for 3 weeks.

1. Write an expression for Jade's total savings.
2. Write an expression for Ben's total savings.
3. Find and simplify the total saved by both of them, and factorise the answer.
4. Find the total when \(x=6\).

Maya expands four double brackets and records the answers in a table.

1. Complete the table.
2. What is the link between the numbers in the brackets and the coefficient of \(x\) and the constant term in the answer?
3. Use your rule to write down the expansion of \((x+4)(x+7)\) without working.
4. Use your rule to predict, then check by expanding, \((x-5)(x+8)\).

Four pupils expand or factorise. One answer is correct for each pupil, but the others have mistakes.

1. Decide whether each is right or wrong.
2. Correct each wrong answer, showing the correct working.
3. Explain Ali's mistake using \(x=1\).
4. Which of the correct answers is a difference of two squares?

A rectangular patio has an area of \((6x^2+9x)\) m\(^2\). One side is \(3x\) metres long.

1. Factorise \(6x^2+9x\) fully.
2. Write the length of the other side.
3. Find the perimeter of the patio in terms of \(x\).
4. Find the dimensions and area when \(x=2\) and check the area.
5. Slabs cost £7.50 per m\(^2\). Find the cost of covering the patio when \(x=2\).

Zara's teacher says: "Think of a number \(n\). Add 4. Square the answer. Subtract 16. Divide by your original number." Zara tries some numbers.

1. Check the first two entries in the table.
2. Write the steps as an algebraic expression.
3. Expand and simplify the top line, then factorise it.
4. Show that the final answer is always \(n+8\).
5. Explain why the trick fails when \(n=0\).

A rectangular picture on a poster is \(x\) cm wide and \((x+4)\) cm tall. There is a border of width 2 cm all the way round.

1. Write expressions for the width and height of the whole poster.
2. Expand the area of the whole poster.
3. Show that the border area is \(8x+32\) and factorise it.
4. Find the border area when \(x=12\), and check with the numbers.
5. Border paper costs 0.5p per cm\(^2\). Find the cost when \(x=12\).

Any odd number can be written as \(2k+1\). The next odd number is \(2k+3\).

1. Expand \((2k+3)^2\) and \((2k+1)^2\).
2. Show that the difference between the squares of two consecutive odd numbers is \(8k+8\).
3. Hence show that the difference is always a multiple of 8.
4. Test your result with \(7^2-5^2\).
5. Use the difference of two squares to work out \(51^2-49^2\), and check it is a multiple of 8.