# 03 factorising, roots, zeros

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Quadratics and Polynomials

Quadratics and PolynomialsPolynomial FactorsFactorisationFactors vs Roots vs Graph

Roots, Factors, Zeros, SolutionsWhat does it mean?!?Remember a graph of a quadratic equation:

Roots, Factors, Zeros, SolutionsWhere the graph line meets the x or the y axes the points of intercept give us solutions!(also called zeros)

Why zeros?!?Any polynomial:

What are factors then?Think of a non-prime number , eg: 6Factors of this number are 2 * 3 = 6So the 2 and 3 are factors of 6, their product makes 6.Two or more numbers (expressions) in a product Zero Factor PropertyThere is a SPECIAL factor law, called:ZERO FACTOR PROPERTYIf a * b = 0 theneither a=0 or b=0or BOTH = 0This will ONLY work for factors of zero(hence the name, doh!)How does this help?Last week we looked at distributive law: (x+2)(x+3) = x2 + 3x + 2x + 2*3 == x2 + 5x + 6

In the first line, I had two expressions:x+2x+3 multiplied. These are my factors!Factors in polynomialsA simple polynomial:

3x + 12Factors in polynomials3 is the a factor which both expressions have, so it was easy to take it out.

What about a bit harder:10x2 15x

FactorsWhat about even harder:

x2y3 + xy3x3 + 6x2 15x2(x y) b(x y)x(x 2) + 3(2 x)Factoring Quadraticsx2 + 4x x 4

x2 5x 6(show splitting b in two numbers)

6x2 13x + 6(show splitting b and finding product)

A tricky one!x2 + 7x 6

How does this factor?!?So, WHY would we factorise?!?Draw a graph of the functions we worked with before:

So, what is the meaning of factors?They are intercepts!!! (zeros!!!)

But why do we call them zeros?

Remember theZERO FACTOR PROPERTY?Zeros (contd)If a * b = 0 thenEither a = 0 orb = 0(or both)If we have two factors, then we can use the above:(x+1)(x-2) = 0HenceZeros (contd)Either:(x+1) = 0 or (x-2) = 0(or both)

So, if x+1 = 0 then x = -1andif x-2 = 0 then x = 2(intercepts!!!)Why roots?Next timePolynomialFactorised

10x2 15x5x(2x-3)

x2y3 + xyxy(xy2 +1)

x2 5x 6x2 6x + 1x 6 = x(x-6) + (x-6) = (x+1)(x-6)

2(x y) b(x y)(2-b)(x-y) - can not draw

x(x 2) + 3(2 x)(x+3)(x-2)

x2 + 4x x 4x(x+4)-(x+4) = (x-1)(x+4)

63x3 - 15x2 18x3x(x2 5x 6)=3x(x+1)(x-6)

6x2 13x + 66x2 9x 4x + 6 = 3x(2x-3)- 2(2x-3) = (3x-2)(2x-3)

x2 + 7x 6???

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