Products and Factors

Year 10 Maths: Chapter 5

In this chapter you will learn to:

  1. The index laws
  2. Fractional indices
  3. Adding and subtracting algebraic fractions
  4. Multiplying and dividing algebraic fractions
  5. Expanding and factorising expressions
  6. Expanding binomial products
  7. Factorising special binomial products
  8. Factorising quadratic expressions

1. The index laws

Write these out on a piece of paper and memorise them!!!
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Exponent rules part 1 | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy

2. Fractional indices

We use fractional indices to work with square, cubes and further roots
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Zero, negative, and fractional exponents | Pre-Algebra | Khan Academy

3. Adding and subtracting algebraic fractions

To add or subtract fractions convert them (if needed) so they have the same denominator and then add or subtract the numerators

See examples on p. 161

Algebraic expression adding fractions | Introduction to algebra | Algebra I | Khan Academy

4. Multiplying and dividing algebraic fractions

  • To multiply fractions, cancel any common factors and multiply the numerator and denominator separately
  • To divide by a fraction (e.g. a/b), multiply by its reciprocal (b/a)
Algebraic expressions with fraction division | Introduction to algebra | Algebra I | Khan Academy

5. Expanding and Factorising Algebraic Expressions

Expanding and factorising are inverse operations.

When 4(2a + 5) is expanded, the answer is 8a + 20

When 8a + 20 is factorised, the answer is 4(a+5)


Summary: Expanding an Expression

Multiply each term inside the brackets by the term outside the brackets:

a(b + c) = ab + bc

a(b - c) = ab - ac


Summary: Factorising an expression

  • Divide the HSC of the terms and write it outside the brackets
  • Divide each term b the HSC and write the answers inside the brackets:

ab + ac = a(b + c)

ab - ac = a(b - c)

  • To check that the factorised answer is correct, expand it

Algebra - expanding and simplifying brackets

6. Expanding binomial products

(a + 3) and (x-2) are called binomial expressions because each expression expression has exactly two terms (binomial means 'two terms').

(a + 3)(x - 2) is called a binomial product because it is a product (multiplication) of two binomial expressions

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Algebra - expanding brackets - binomials

Perfect squares

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Algebra - Perfect Square Factoring and Square Root Property

Difference of two squares

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Factor the Difference of Two Squares

8. Factorising quadratic expressions

  • A quadratic expression is an algebraic expression where the highest power of the variable is '2'.
  • Such an expression is called a 'trinomial' because it has three terms

To factorise such terms:

  • Find two numbers that have a sum of b and a product of c.
  • Use these two numbers to write a binomial product in the form (x ____)(x_____)

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❤² How to Solve Quadratic Equations By Factoring (mathbff)