Simplifying, expanding and factorising

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Apuntes de la lección

Collecting Like Terms

Like terms have the same variable(s) raised to the same power; only the coefficients differ. For example, 3x and 5x are like terms, but 3x and 3x² are not.
To simplify, add or subtract the coefficients of like terms. Keep the variable part unchanged: 4a + 7a = 11a, and 9b - 2b = 7b.
Watch signs carefully: when subtracting, distribute the minus sign to every term in the bracket. For example, (3x + 2) - (x - 5) = 3x + 2 - x + 5 = 2x + 7.
Terms with different variables or powers cannot be combined: 2x + 3y stays as 2x + 3y, and x² + x stays as x² + x.
Always simplify fully: 5x + 3 - 2x + 4 = 3x + 7.

Multiplying a Single Term over a Bracket

Use the distributive law: a(b + c) = ab + ac. Multiply the term outside by each term inside the bracket.
Example: 3(2x + 5) = 6x + 15.
Be careful with negatives: -2(3x - 4) = -6x + 8. The negative multiplies every term inside.
When the outside term is a variable, multiply coefficients and add powers: 2x(3x² + 4) = 6x³ + 8x.
Always simplify the result by collecting like terms if possible.

Expanding Products of Binomials

Use FOIL (First, Outer, Inner, Last) or the distributive law twice to expand (a + b)(c + d) = ac + ad + bc + bd.
Example: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6.
Common error: (a + b)² is NOT a² + b². Instead, (a + b)² = a² + 2ab + b².
For three binomials, expand two first, then multiply the result by the third: (x + 1)(x + 2)(x + 3) = (x² + 3x + 2)(x + 3) = x³ + 6x² + 11x + 6.
Always simplify by collecting like terms after expanding.

Factorising by Taking Out Common Factors

Factorising is the reverse of expanding: write an expression as a product of factors.
Identify the highest common factor (HCF) of all terms. For example, 6x² + 9x = 3x(2x + 3).
Take out the HCF completely: 12x³ - 8x² = 4x²(3x - 2).
Common error: taking out only part of the common factor. Always take out the full HCF.
Check your answer by expanding the bracket to see if you get the original expression.

Factorising Quadratics of the Form x^2 + bx + c

A quadratic x² + bx + c factorises as (x + p)(x + q) where p + q = b and pq = c.
Find two numbers that multiply to give c and add to give b. For x² + 5x + 6, the numbers are 2 and 3, so (x + 2)(x + 3).
If c is positive, both p and q have the same sign as b. If c is negative, one is positive and one is negative.
Example: x² - x - 6 = (x - 3)(x + 2) because -3 + 2 = -1 and -3 × 2 = -6.
Always check by expanding your factorisation.

Difference of Two Squares

The difference of two squares is a special pattern: a² - b² = (a + b)(a - b).
Recognise it when you have a squared term minus another squared term. For example, x² - 9 = (x + 3)(x - 3).
Also applies to expressions like 4x² - 25 = (2x + 5)(2x - 5).
Note: a sum of squares, a² + b², does not factorise over real numbers.
Use this pattern to factorise quickly and to simplify algebraic fractions.

Factorising Quadratics with Coefficient of x^2 Not 1 (Higher)

For ax² + bx + c with a > 1, use the ac method or trial and error.
Multiply a and c to get ac. Find two numbers that multiply to ac and add to b.
Rewrite the middle term using those two numbers, then factor by grouping. Example: 2x² + 7x + 3: ac = 6, numbers 6 and 1, so 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Alternatively, use trial and error: (2x + 1)(x + 3) works because 2x × 3 + 1 × x = 7x.
Always check by expanding.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, factorise the numerator and denominator completely, then cancel common factors.
Example: \frac{x² - 9}{x² + 5x + 6} = \frac{(x + 3)(x - 3)}{(x + 3)(x + 2)} = \frac{x - 3}{x + 2}.
Common error: cancelling individual terms instead of factors. You can only cancel factors that are multiplied, not terms that are added or subtracted.
Retain every original denominator restriction. In (x² − 9)/(x² + 5x + 6), x must differ from both −3 and −2, even after cancelling x + 3.
Always factorise fully before cancelling.

Common Misconceptions and Errors

Failing to distribute a negative across every term in a bracket: -2(x - 3) = -2x + 6, not -2x - 3.
Believing (a + b)² = a² + b²: the correct expansion is a² + 2ab + b².
Taking out only part of the common factor: always take the highest common factor.
Cancelling terms in algebraic fractions: cancel factors, not terms. For example, \frac{x + 2}{x} cannot be simplified to 2.
Losing solutions by cancelling factors that could be zero: when solving equations, don't cancel a variable factor unless you consider the case where it equals zero.

Typical Question Forms

Expand and simplify products of brackets, e.g., (x + 4)(x - 1).
Factorise fully a quadratic expression, e.g., x² - 7x + 12.
Simplify an algebraic fraction, e.g., \frac{x² - 4}{x² - 4x + 4}.
Show that two expressions are equivalent by simplifying or expanding both.
Write and simplify an expression for the area or perimeter of a shape with algebraic side lengths.

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Preguntas de práctica

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1.Which of the following is the correct definition of 'like terms'?
Easy
- ATerms that have the same variable(s) raised to the same power(s)
- BTerms that have the same numerical coefficient
- CTerms that have the same variable(s) regardless of power
- DTerms that are equal in value
2.The expression 4x + 3y - 2x + y simplifies to 2x + 4y.
Easy
True or false?
3.Simplify: 7a + 3b - 2a + 5b
Easy
- A5a + 8b
- B9a + 8b
- C5a - 2b
- D9a - 2b
4.Expand and simplify: (x + 3)(x + 5)
Medium
- Ax² + 8x + 15
- Bx² + 15x + 8
- Cx² + 2x + 15
- Dx² + 8x + 8
5.Factorise completely: 6x² + 9x
Medium
- A3x(2x + 3)
- B3(2x² + 3x)
- Cx(6x + 9)
- D3x(2x² + 3)
6.Which of the following are equivalent to x² - 9? (select all that apply)
Medium
- A(x - 3)(x + 3)
- B(x - 3)²
- Cx² - 6x + 9
- D(x + 3)(x - 3)
- Ex(x - 9)
7.Match each expression with its simplified form.
Medium
- 2x + 3x
- 2x × 3x
- 2x² + 3x²
- 5x
- 6x²
- 5x²
8.Arrange the steps to factorise x² + 7x + 12 in the correct order.
Medium
- Find two numbers that multiply to 12 and add to 7
- Write the factorised form: (x + 3)(x + 4)
- Identify the quadratic is of the form x² + bx + c
- Check the numbers 3 and 4 satisfy the conditions

Más preguntas

9.Arrange the steps to expand and simplify (x + 3)(x + 2) in the correct order.
Medium
- Expand each product: x² + 2x + 3x + 6
- Collect like terms: x² + 5x + 6
- Write the two binomials in a grid or use FOIL
- Identify the terms: x², 2x, 3x, 6
10.Which forms are equivalent to (x + 2)²? Select all that apply.
Medium
- Ax² + 4
- Bx² + 4x + 4
- C(x + 2)(x + 2)
- Dx² + 2x + 2x + 4
- Ex² + 2x + 4
11.For every real a and b, (a + b)² is equal to a² + b².
Medium
True or false?
12.Factorise x² - 9.
Medium
- A(x - 3)(x + 3)
- B(x - 3)²
- C(x + 3)²
- D(x - 9)(x + 1)
13.Which of the following are equivalent to \frac{x² - 1}{x - 1} for x ≠ 1? (select all that apply)
Hard
- Ax + 1
- Bx - 1
- C1
- Dx + 1 (if x ≠ 1)
- Ex - 1 (if x ≠ 1)
14.A rectangle has side lengths (2x + 3) cm and (x − 1) cm, with x > 1. Write a simplified expression for its area in cm².
Medium
- A2x² + x - 3
- B2x² + 5x - 3
- C2x² + x + 3
- D2x² - x - 3
15.Simplify: 5(2x - 3) - 2(3x + 1)
Medium
- A4x - 17
- B4x - 13
- C16x - 17
- D4x - 15
16.When simplifying 3(x + 2) - (x + 1), the result is 2x + 5.
Medium
True or false?
17.Simplify: 5x - 3y + 2x + 7y
Easy
- A7x + 4y
- B3x + 10y
- C7x - 4y
- D3x - 4y
18.Expand and simplify: 3(2x + 1) - 2(x - 4)
Medium
- A4x + 11
- B4x - 5
- C8x - 5
- D4x + 5
19.Factorise completely: 12x² - 8x
Medium
- A4x(3x - 2)
- B2x(6x - 4)
- C4(3x² - 2x)
- Dx(12x - 8)
20.Which of the following is the correct expansion of (x - 3)(x + 5)?
Medium
- Ax² + 2x - 15
- Bx² - 2x - 15
- Cx² + 8x - 15
- Dx² - 8x - 15
21.Which of the following are factors of x² - 4x - 12? (select all that apply)
Medium
- A(x + 2)
- B(x - 2)
- C(x - 6)
- D(x + 6)
- E(x + 3)
22.The expression (x - 2)² is equivalent to x² - 4x + 4.
Medium
True or false?
23.Arrange the steps to factorise x² + 5x - 14 in the correct order.
Medium
- Find two numbers that multiply to -14 and add to 5.
- Write the factors as (x + 7)(x - 2).
- Check by expanding: (x+7)(x-2) = x² + 5x - 14.
- Identify the numbers as 7 and -2.
24.A rectangle has length (2x + 3) cm and width (x − 2) cm, with area 30 cm². Which fully simplified quadratic equation, with right side zero, gives x?
Hard
- A2x² - x - 36 = 0
- B2x² - x - 6 = 30
- C2x² + 7x - 6 = 30
25.Which of the following expressions are equivalent to (x + 3)² - (x - 1)²? (select all that apply)
Hard
- A8x + 8
- B4x + 8
- C8(x + 1)
- D4(2x + 2)
- E2x + 8
26.The expression (x + 2)(x - 3) simplifies to x² - x - 6.
Medium
True or false?
27.Which is the correct expansion of (x - 4)(x + 2)?
Medium
- Ax² - 2x - 8
- Bx² - 2x + 8
- Cx² + 2x - 8
- Dx² - 6x - 8
28.Arrange the steps to factorise x² + 3x - 10.
Medium
- Find two numbers that multiply to -10 and add to 3
- Write the factors as (x + 5)(x - 2)
- Identify the numbers 5 and -2
- Check by expanding: (x+5)(x-2) = x² +3x -10
29.The area of a rectangle is (x² + 5x + 6) cm² and its length is (x + 2) cm. Which expression represents its width?
Hard
- Ax + 3
- Bx + 4
- Cx - 3
- Dx - 4
30.In algebra, 'like terms' are terms that have the same variable raised to the same power.
Easy
True or false?
31.Which of the following is the correct expansion of 3(x + 4)?
Easy
- A3x + 4
- B3x + 12
- Cx + 12
- D3x + 7
32.Which of the following is the correct expansion of (x + 3)(x + 5)?
Easy
- Ax² + 8x + 15
- Bx² + 15x + 8
- Cx² + 8x + 8
- Dx² + 15x + 15
33.Which of the following is the correct factorisation of x² - 5x + 6?
Medium
- A(x - 2)(x - 3)
- B(x + 2)(x + 3)
- C(x - 1)(x - 6)
- D(x + 1)(x - 6)
34.When simplifying \frac{x² - 1}{x - 1} for x ≠ 1, it is valid to cancel the x terms in the numerator and denominator.
Hard
True or false?
35.Match each expression to its simplified form.
Medium
- 3x + 2x
- 5x - 2x
- 2x + 3x - x
- 5x
- 3x
- 4x
36.Using FOIL order (First, Outer, Inner, Last), arrange the expansion steps for (x + 4)(x − 2), then collect like terms.
Medium
- Multiply x by x to get x²
- Multiply x by -2 to get -2x
- Multiply 4 by x to get 4x
- Multiply 4 by -2 to get -8
- Combine x², -2x, 4x, and -8 to get x² + 2x - 8
37.Which of the following are correct expansions of (2x + 3)(x - 1)? (select all that apply)
Medium
- A2x² + x - 3
- B2x² - 2x - 3
- C2x² - 3x - 3
- D2x² - 2x + 3x - 3
38.When expanding 3(x - 2), the result is 3x - 2.
Easy
True or false?
39.Factorise completely: 8x² - 12x
Medium
- A4x(2x - 3)
- B2x(4x - 6)
- C4(2x² - 3x)
- Dx(8x - 12)
40.Which of the following is the correct factorisation of 2x² + 7x + 3?
Hard
- A(2x + 1)(x + 3)
- B(2x + 3)(x + 1)
- C(2x - 1)(x - 3)
- D(2x - 3)(x - 1)
41.Fill in the blank to complete the expansion: 3(x + 4) = 3x + ___
Easy
3(x + 4) = 3x + ____
42.Solve for x: x² - 5x + 6 = 0. Enter the larger root.
Medium
43.Simplify the algebraic fraction: \frac{x² - 4}{x - 2} (x ≠ 2).
Hard
44.Complete the expansion: 2x(3x - 5) = ___ - ___
Medium
2x(3x - 5) = ____ - ____
45.Factorise x² + 7x + 12 into (x + a)(x + b), with a < b. What is a?
Medium
46.A rectangle has sides (2x + 3) cm and (x − 1) cm, with x > 1, and area 42 cm². Find x.
Medium
47.Simplify the expression: 4(2x - 3) - 3(x + 1)
Medium
48.Factorise x² − 10x + 21 into (x − a)(x − b), with 0 < a < b. What is a?
Medium
49.Complete the expansion: 5x(2x + 3) = ___ + ___
Easy
5x(2x + 3) = ____ + ____
50.Factorise the expression by taking out the highest common factor: 9x³ - 6x² = ___ (___ - ___)
Medium
9x³ - 6x² = ____ (____ - ____)
51.Simplify: 3x + 5 - 2x + 7. Enter the coefficient of x.
Easy
52.Complete the expansion: 4(2x - 3) = ___ x - ___
Medium
4(2x - 3) = ____ x - ____
53.Show that (x + 2)² - 4 is equivalent to x(x + 4).
Hard
54.Show that (x + 3)² - 9 is equivalent to x(x + 6).
Hard
55.Show that (x + 1)² - 1 is equivalent to x(x + 2).
Hard
56.Explain why \frac{x² - 4}{x - 2} simplifies to x + 2 for x ≠ 2, but you must not cancel individual terms.
Hard
57.Solve for x: x² - 5x + 6 = 0. Enter the larger root.
Medium
- A4
- B2
- C5
- D3
58.Factorise x² + 7x + 12 into (x + a)(x + b), with a < b. What is a?
Medium
- A4
- B5
- C2
- D3
59.A rectangle has sides (2x + 3) cm and (x − 1) cm, with x > 1, and area 42 cm². Find x.
Medium
- A4.5 cm
- B7 cm
- C6 cm
- D4 cm
60.Factorise x² − 10x + 21 into (x − a)(x − b), with 0 < a < b. What is a?
Medium
- A2
- B5
- C4
- D3
61.Simplify: 3x + 5 - 2x + 7. Enter the coefficient of x.
Easy
- A3
- B0
- C2
- D1

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