Algebraic Techniques: Expanding, Factoring, Fractions, and Quadratics

Expanding binomial products

• Uses the distributive law.
• General rule: (a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd
• Diagrammatic interpretation: (a+b)(c+d)=ac+ad+bc+bd
• Example:
  (x+1)(x+5)=x^2+5x+x+5= x^2+6x+5

Perfect squares and the difference of two squares

• Perfect squares: examples include 3^2=9,
  otag a^2,
  otag (2y)^2,
  otag (x-1)^2,
  otag (3-2y)^2
• Expanding perfect squares:
  (a+b)^2=(a+b)(a+b)=a^2+ab+ba+b^2=a^2+2ab+b^2
• Difference of two squares: (a+b)(a-b)=a^2-ab+ba-b^2 = a^2-b^2
  • Also, (a-b)(a+b)=a^2-b^2
  • The result is a difference of two squares: a^2-b^2=(a-b)(a+b)

Factorising algebraic expressions (HCF and beyond)

• Factorising with a common factor (HCF): take out the highest common factor.
  • Could be a number: 2x+10=2(x+5)
  • Could be a pronumeral (variable): x^2+5x=x(x+5)
  • Product of numbers and pronumerals: 2x^2+10x=2x(x+5)
• A factorised expression can be checked by expansion: 2x(x+5)=2x^2+10x

Factorising the difference of two squares (Ex 8D)

• Rule: a^2-b^2=(a+b)(a-b)
• Examples:
  • x^2-16=(x+4)(x-4)
  • 9x^2-100=(3x)^2-10^2=(3x+10)(3x-10)
  • 25-4y^2=5^2-(2y)^2=(5+2y)(5-2y)
• First take out common factors when possible: 2x^2-18=2(x^2-9)=2(x+3)(x-3)

Factorising by grouping in pairs (Ex 8E)

• Method: group four-term expressions into two pairs, factor each pair, then extract a common binomial factor.
• Steps:
  • Rearrange terms if needed to reveal a common factor.
  • Factorise each pair separately.
  • Take out the common binomial factor.
• Examples:
  • x^2+3x-2x-6 = x(x+3)-2(x+3) = (x+3)(x-2)
  • This could also be written as
    x^2+5x+6
    ightarrow x^2+2x+3x+6 = x(x+2)+3(x+2)=(x+2)(x+3)
  • x^2-2x-15
    ightarrow x(x+3)-5(x+3)=(x+3)(x-5)

Factorising monic quadratic trinomials (Ex 8F)

• Form: x^2+bx+c where the leading coefficient is 1 (monic).
• Method: find two numbers that multiply to $c$ and add to $b$; factor as (x+m)(x+n)=x^2+(m+n)x+mn
• Check by expansion: x^2+bx+c=(x+m)(x+n) ext{ if } m+n=b ext{ and } mn=c
• Examples:
  • x^2-3x-10=(x-5)(x+2) (since $-5+2=-3$ and $(-5)(2)=-10$)
  • If $c<0$, the signs in the factors differ (one positive, one negative).
    Example: x^2+5x-6=(x+6)(x-1) (since $6
    earrow$ and $-1$ multiply to $-6$ and sum to $5$)
  • If $c>0$, the signs in the factors are the same.
    Example: x^2+5x+6=(x+2)(x+3)
    Example: x^2-5x+6=(x-2)(x-3)

Cross Method (Ex 8D or 8F style technique for ax^2+bx+c)

• Used to factorise quadratics of the form ax^2+bx+c without factoring by inspection.
• Idea: factorise $ax^2$ and $c$; find a pair of such factors that combine to yield the middle term $bx$ when cross-multiplied.
• Example: Factorise 6x^2-x-15
  • Factors of 6x^2: $(x,6x)$, $(2x,3x)$
  • Factors of -15: $(15,-1)$, $(-15,1)$, $(5,-3)$, $(-5,3)$
  • Choose a pair that gives the middle term after cross-multiplication. The correct choice is:
    (2x+3)(3x-5)
  • Verification by expansion: (2x+3)(3x-5)=6x^2-10x+9x-15=6x^2-x-15
• Process may require trying several factor pairs until the sum matches $bx$.

Simplifying algebraic fractions: multiplication and division (Ex 8H)

• Principles:
  • Factorise expressions where possible.
  • Cancel any common factors before multiplying.
  • Multiply numerators together and denominators together after cancellation.
• Example illustrating common-factor cancellation (incorrect vs correct):
  • Incorrect: \frac{2x+4}{2\cdot 21}=\frac{2x+4}{42} (not simplified)
  • Correct: factorise first: \frac{2(x+2)}{2\cdot 21}=\frac{x+2}{21}
• Multiplication rule:
  • If you have \frac{A}{B}\cdot\frac{C}{D}, then after simplifying common factors, you get \frac{AC}{BD}.
• Division rule (reciprocal):
  • To divide by a fraction, multiply by its reciprocal: \frac{A}{B} \div \frac{C}{D}=\frac{A}{B}\cdot\frac{D}{C}, provided $C\neq0$ and all fractions are simplified first.

Simplifying algebraic fractions: addition and subtraction (Ex 8I)

• Steps to combine fractions:
  • Determine the least common denominator (LCD).
  • Express each fraction with the LCD as its denominator.
  • Add or subtract the numerators.
• Important notes:
  • The product of two numbers of opposite sign is negative.
  • The product of two negative numbers is positive.
• Example framework (non-specified numbers):
  • Combine \frac{A}{x+3} and \frac{B}{x-1} using LCD $(x+3)(x-1)$:
    \frac{A}{x+3}+\frac{B}{x-1}=\frac{A(x-1)+B(x+3)}{(x+3)(x-1)}
• Demonstration that a common denominator can be a product of two algebraic expressions.
  • Example form: \frac{3}{x+3}+\frac{2}{x-1}=\frac{3(x-1)+2(x+3)}{(x+3)(x-1)}

Further addition and subtraction of algebraic fractions (Ex 8J)

• When subtraction signs are involved, apply sign rules carefully:
  • Product of opposite signs is negative; product of two negatives is positive.
• Examples show common denominators can be products of binomials, e.g.:
  • Combine \frac{5x+7}{x+3} - \frac{2x-1}{x-1} by using LCD $(x+3)(x-1)$:
    \frac{(5x+7)(x-1) - (2x-1)(x+3)}{(x+3)(x-1)}
• Another example structure:
  • \frac{3+(1)}{x+3}+\frac{2(x+3)}{x-1} \rightarrow \text{LCD}=(x+3)(x-1)

Equations involving algebraic fractions (Ex 8K)

• Strategy for equations with more than one fraction:
  • Multiply every term on both sides by the LCD to clear fractions.
  • Then simplify the resulting equation using standard algebra.
  • Alternatively, express each fraction with the same denominator and simplify by adding/subtracting.
• Example outline (general procedure):
  • Given an equation with fractions on both sides, multiply by the LCD to obtain a polynomial equation.
  • Solve the resulting equation using previously learned methods (factoring, quadratic formula, etc.).
  • Check that solutions satisfy the original equation (no extraneous roots from multiplying by zero).

Quadratic equations and graphs of parabolas (Chapter 10)

• Quadratic equations in standard form:
  ax^2+bx+c=0 \text{ with } a\neq 0
• The Null Factor Law: if $pq=0$ then $p=0$ or $q=0$.
  • Example: if $(x+1)(x-3)=0$, then $x=-1$ or $x=3$.
  • Relationship: the linear factors are roots of the quadratic.
• Factorising example:
  • $x^2-2x-3=(x+1)(x-3)$ because the roots are $-1$ and $3$.
• Move all terms to the left-hand side when solving from factored form.

Solving ax^2 + bx = 0 and x^2 - d^2 = 0 by factorising (Ex 10B)

• For equations of the form ax^2+bx=0
  • Factor out $x$: x(ax+b)=0
  • Solutions: x=0 \text{ or } ax+b=0\Rightarrow x=-\frac{b}{a}
• For equations of the form ax^2=d
  • Divide by $a$, then take square roots: x^2=\frac{d}{a}\Rightarrow x=\pm\sqrt{\frac{d}{a}}
• For equations of the form x^2-d^2=0
  • Factor: (x-d)(x+d)=0
  • Solutions: x=\pm d
• Example set (from Ex 10B):
  • $x^2+4x=0 \Rightarrow x(x+4)=0 \Rightarrow x=0$ or $x=-4$ (depending on sign convention)
  • $2x^2-8x=0 \Rightarrow 2x(x-4)=0 \Rightarrow x=0$ or $x=4$
  • $x^2=4x \Rightarrow x^2-4x=0 \Rightarrow x(x-4)=0 \Rightarrow x=0$ or $x=4$
  • $x^2-4=0 \Rightarrow (x+2)(x-2)=0 \Rightarrow x=\pm 2$

Solving quadratic equations by factoring (Ex 10C)

• For $x^2+bx+c=0$ (monic quadratic), find two numbers that multiply to $c$ and add to $b$.
• Then express as a product: x^2+bx+c=(x+m)(x+n) with $m+n=b$ and $mn=c$.
• Use Null Factor Law to read off roots: x=-m\quad \text{or}\quad x=-n
• Examples:
  • $x^2-3x-28=0 \Rightarrow (x-7)(x+4)=0 \Rightarrow x=7, -4$
  • $x^2-6x+9=0 \Rightarrow (x-3)^2=0 \Rightarrow x=3$
• Important note: If the coefficient of $x^2$ is 1 (monic), the factorisation method directly uses two integers that sum to $b$ and multiply to $c$.

Using quadratic equations to solve problems (Ex 10D)

• Steps:
  • Define the variable (e.g., let $x$ be a length/breadth).
  • Relate quantities with an equation (often area, perimeter, or other geometric relation).
  • Solve the resulting quadratic (by factoring or other methods).
  • Check feasibility (dimensions must be positive, etc.).
  • Report the answer in words and verify reasonableness.
• Example problem (book face dimensions):
  • Let $x$ be the breadth in cm; length is $x+4$ cm; area is $320$ cm$^2$.
  • Equation: x(x+4)=320
  • Rearrange: x^2+4x-320=0
  • Factor: (x-16)(x+20)=0
  • Solutions: x=16,\; x=-20
  • Feasible solution: $x=16$ (breadth positive interpreted). Then length $= x+4 = 20$ cm.

Quick reference: key formulas and laws

• Distributive law for binomials: (a+b)(c+d)=ac+ad+bc+bd
• Perfect square expansion: (a+b)^2=a^2+2ab+b^2
• Difference of squares: a^2-b^2=(a+b)(a-b)
• Factor out greatest common factor (HCF) from terms.
• Factorising by grouping in pairs: group terms to reveal a common binomial factor.
• Monic quadratic factorisation: find $m,n$ with $m+n=b$ and $mn=c$, then x^2+bx+c=(x+m)(x+n)
• Cross method: a method to obtain the required splitting of the middle term by using factor pairs of $ax^2$ and $c$.
• Null Factor Law: if pq=0 then p=0 ext{ or } q=0
• Quadratic equations in standard form: ax^2+bx+c=0, \, a\neq 0$$
• Solving linear-quadratic mixtures: factor out common factors, or restructure as a difference of squares when applicable.
• Algebraic fractions: rules for multiplication, division, addition, and subtraction with full simplification via factorisation and cancellation.
• Clearing fractions in equations: multiply both sides by the LCD to obtain a polynomial equation to solve.
• Graphs of parabolas: quadratic equations correspond to parabolas; solutions are x-intercepts (roots).