Algebraic factors

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ITEM	TYPE	NUMBER
Factors and quadratics	Workout	71 slides
Algebraic factors	Library	-1 questions

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SAMPLE FROM THE WORKOUT

Showing American English version

SLIDE 1 - QUESTION 1

SLIDE 2 - SOLUTION

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SAMPLE FROM THE LIBRARY

Showing American English version

QUESTION [difficulty 0.1]

SOLUTION

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DEPENDENCIES

		282: Algebraic fractions

		286: Algebraic factors

		296: Transpositions
		298: Patterns and further sequences

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CONCEPTS

ITEM

LEV.

Factorising when the common term is bracketed

656.3

Commutativity

656.5

Expansion of binomial products

656.8

Factorising to binomal product

657.3

Quadratic factorisation, a = 1, positive integer solutions

657.7

Quadratic factorisation, a = 1, negative coefficients

658.0

Expanding algebraic squares

658.5

Finding coefficient in an algebraic square

658.9

Factorising an algebraic square

659.3

Expanding factors in difference of two squares

659.5

Difference of two squares

659.7

Graphical meaning of difference of two squares

659.8

Factorising difference of two squares

659.9

Factors of zero

660.1

Zero divisor

660.3

Solving quadratic equations for roots

660.3

The Greek alphabet in mathematics

661.0

Expand and collect (x – α)(x – β)

661.2

Consolidation of properties of the quadratic function

661.3

Relationship between graph of parabola and roots

661.6

Making a sketch of a parabola from the roots

661.9

Factorsing quadratic when a = –1

662.7

Effect of negative leading coefficient on parabola

662.9

RAW CONTENT OF THE WORKOUT

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SLIDE 1 Consolidation ? Expand and collect ? Expand and collect ? Solve ? Factorise ? Factorise ? Solve SLIDE 2 ? ? ? ? ? ? SLIDE 3 Factorising when the common term is bracketed Factorise ? ? ? ? ? ? SLIDE 4 ? ? ? ? ? ? SLIDE 5 Commutativity The order in which numbers or terms are multiplied makes no difference. You can multiply numbers and terms in any order. Brackets can appear in any order SLIDE 6 Exercise in commutativity Write down every possible way in each of the following may appear. (Do not expand any bracket.) ? ? ? ? SLIDE 7 ? Recall also that we use dots and crosses for multiplication, as well as just putting letters side by side. ? ? ? They are all the same. SLIDE 8 Binomial products The word binomial indicates two of something. This is a binomial product because (a) there are two expressions, which are in brackets, and (b) they are multiplied (product) together. Example Expand and collect Rule. Every term in the first bracket is multiplied once by every term in the second bracket. Work systematically. Start with the first term in the first bracket and work your way systematically down all the terms in the second bracket. When that is done, go to the second term in the first bracket and work your way through the second bracket again. SLIDE 9 Expand and collect ? ? ? ? ? ? SLIDE 10 ? ? ? ? ? ? SLIDE 11 Expand and collect ? ? ? ? ? ? SLIDE 12 ? ? ? ? ? ? SLIDE 13 Example Factorise ? ? ? ? SLIDE 14 ? ? ? ? SLIDE 15 Find all the pairs of numbers whose products are 6, 9, 12, 14, 15 and 28. SLIDE 16 Number Product factors 6 9 12 14 15 28 SLIDE 17 Example Factorise Solution We are looking for a binomial product that when expanded gives . That is two expressions in brackets. ? Since the leading term in is , the first term in each bracket must be x ? The numbers must be factors of 6. There are only two possibilities: or . ? The middle term is 7 and is found by adding the factors of 6. Since we try 1 and 6 in the bracket. ? We check to see if this is correct by multiplying the binomial product. This is indeed the right answer � we know we got it right. SLIDE 18 Factorise ? ? ? ? SLIDE 19 ? ? ? ? SLIDE 20 Example Factorise Solution There is a negative sign in front of the 6. We must now consider all the positive and negative factors of , and their corresponding sums. Of these we see that the option that is correct is the last as we a require for the middle term. ? ? ? ? Check. ? SLIDE 21 Factorise ? ? ? ? SLIDE 22 ? ? ? ? SLIDE 23 Factorise ? ? ? ? SLIDE 24 ? ? ? ? SLIDE 25 Example Expand and simplify ? ? ? ? ? ? SLIDE 26 ? ? ? ? ? ? SLIDE 27 Expand and simplify ? ? SLIDE 28 ? ? SLIDE 29 ? If , find a. ? If , find a. SLIDE 30 ? If , find a. Solution ? If , find a. Solution SLIDE 31 Complete the above diagram. SLIDE 32 SLIDE 33 Factorise ? ? ? ? SLIDE 34 ? ? ? ? SLIDE 35 Expand and simplify ? ? ? SLIDE 36 ? ? ? SLIDE 37 The difference of two squares The expression is called the difference of two squares. A square has been removed from a larger square . Factorisation When is expanded the middle terms cancel out and disappear. SLIDE 38 Graphical interpretation of the difference of two squares SLIDE 39 Factorise ? ? ? ? ? ? SLIDE 40 ? ? ? ? ? ? SLIDE 41 Factors of zero Fill in the missing gaps ? ? ? SLIDE 42 ? ? ? SLIDE 43 Zero divisor If the product of two numbers is zero, then one of those numbers (or both) must be zero. Solving quadratic equations Solve Solution ? ? ? SLIDE 44 Solve ? ? ? ? ? ? ? ? SLIDE 45 ? ? ? ? ? ? ? ? SLIDE 46 Consolidation x -4 -3 -2 -1 0 1 2 3 4 16 Complete the table and plot the graph of SLIDE 47 x -4 -3 -2 -1 0 1 2 3 4 16 9 4 1 0 1 2 9 16 SLIDE 48 x -4 -3 -2 -1 0 1 2 3 4 -5 -2 10 Complete the table and plot the graph of SLIDE 49 x -4 -3 -2 -1 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 5 6 10 4 0 -2 -2 0 4 10 18 SLIDE 50 Greek alphabet Greek letters are often used in mathematics. The first four Greek letters are SLIDE 51 Expand and collect . SLIDE 52 SLIDE 53 Properties of the quadratic function The quadratic function is The graph of is a parabola. The factors of are and . The roots are . The roots are the solutions to the linear equations, and . The parabola is symmetric about a vertical line of reflection symmetry, also called its axis of symmetry. The parabola has a turning point. This is either a minimum or a maximum. The roots are found where the parabola cuts the x-axis. SLIDE 54 This is the graph of the quadratic function Draw onto the graph the axis of symmetry. Mark the roots and the minimum point. SLIDE 55 SLIDE 56 If the parabola does not cut the x-axis, then the quadratic function does not have any roots and cannot be factorised (with real numbers). If the parabola touches the x-axis, then it has one root and one factor. In the second case, we sometimes say the quadratic function has one root repeated and one factor repeated, because and . SLIDE 57 The diagram shows the graph of . Draw onto the graph the axis of symmetry, the minimum and the root(s). SLIDE 58 Axis of symmetry, Minimum and root SLIDE 59 Making a sketch for a quadratic function If a quadratic function can be factorised, then a sketch of its graph, a parabola, can be found as follows. ? Factorise the quadratic to ? The roots are ? Find the average of . Then the axis of symmetry is ? Substitute into to find the minimum (or maximum) value of y SLIDE 60 Example Sketch the quadratic Solution SLIDE 61 Sketch the quadratic SLIDE 62 SLIDE 63 Sketch the quadratic SLIDE 64 The sketch also includes the additional information that the intercept on the y-axis is at . This is because when , . SLIDE 65 Sketch the quadratic SLIDE 66 SLIDE 67 When the leading coefficient is negative Example Factorise Solution Factorise ? ? SLIDE 68 ? ? SLIDE 69 The effect of a leading negative coefficient The graph of is the reflection of the graph of in the x-axis. Whereas has a minimum, has a maximum. SLIDE 70 Sketch the quadratic SLIDE 71