Factor polynomials and simplify expressions Factor the polyn

Factor polynomials and simplify expressions

Factor the polynomials 10x^2 + 29x + 21 18x^2-50y^2 x^2 + 14 x + 49-25y^2 simplify the expressions 5a^2 + 22a + 21/a^4 - 81 - 25a62 + 70a+ 49/a^2 - 3a 25x/3x - 25 + 125/3x^2 - 25x + 5/x

Solution

2 a) 10x^2 + 29x +21

Using the qudaratic root from formula and write it as factors:

x = ( -29 + / - sqrt( 29^2 - 840) )/20 = ( -29 + /- 1)/20

x = - 3/2 ; x = -28/20 = - -7/5

10x^2 + 29x +21 = (2x +3)( 5x +7)

b) 18x^2 - 50y^2

Taking out 2 as common: 2( 9x^2 -25y^2)

Using the standard formula: a^2 - b^2 = ( a- b)( a+b)

2( 9x^2 -25y^2) = 2( (3x)^2 - (5y)^2 ) = 2 { ( 3x -5y)( 3x +5y) }

3. a) ( 5a^2 + 22a +21)/(a^4 - 81) / ( 25a^2+70a +49)/( a^2 -3a)

factorise ( 5a^2 + 22a +21) . Find roots using quadratic equation root formula

a = ( -22 +/ - sqrt(64) )/10 ; a= -3 ; x= -7/5

So, factors of 5a^2 + 22a +21 = (x+3)(5x+7)

factorise   25a^2+70a +49 . Find roots using quadratic equation root formula

x = ( -70 + / - sqrt( 0) )/50 ; x= - 7/5 ; x= -7/5

factors of   25a^2+70a +49 = (5x +7)^2

So, further ( a^4 -81) =(a^2 -9)( a^2 +9) = (a+3)(a-3)(a^2+9)

( 5a^2 + 22a +21)/(a^4 - 81) / ( 25a^2+70a +49)/( a^2 -3a)

(a+3)(5a+7)/(a+3)(a-3)(a^2+9) / ( 5a+7)^2(a+3)(a-3)

= (a+3)(5a +7)(a+3)(a-3)/(5a+7)^2(a+3)(a -3)( a^2 +9)

= (a+3)/(5a +7)(a^2 +9)