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Solving Polynomials by factorizing

Factor

1. x^2 + 12x +36 - y^6

2.9y^2 + 12y + 4 - x^2

solve each equation

n^2 + n = 72

(4x + 9)(x - 4)(x + 1) = 0

m^3 = m^2 + 12m

(x + 4)(5x - 1 ) = 0

x^ + 6x - 7 = 0

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The longer leg of a right triangle is 4 feet longer than the other leg. Find the length of the two legs if the hypotenuse is 20 feet.

After t seconds, the height h(t) of a model rocket lauched from the ground into the air is given by the function h(t) = -16t^2 + 80t. Find how long it takes the rocket to reach a height of 96 feet.

The function W (x) = 0.5x^2 gives the number of servings of wedding cake that can be obtained from a two-layer x-inch square wedding cake tier. What size square wedding cake tier is needed to serve 50 people?

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30.x^2 + 12x +36 - y^6
first we can factorize
x^2+12x+36
On splitting middle term we get
x^2+6x+6x+36
=x(x+6)+6(x+6)
=(x+6)*(x+6)
Now put this value in original polynomial
x^2 + 12x +36 - y^6
=(x+6)^2-y^6
y^6 may be written as (y^3)^2
we get
=(x+6)^2-(y^3)^2

We know that (a^2-b^2) =(a+b)(a-b)
Comparing above relation with our polynomial we can write
(x+6)^2-(y^3)^2 = {(x+6)+y^3}*{(x+6)-y^3}=(x+6-y^3)(x+6+y^3)

34.9y^2 + 12y + 4 - x^2
First we can factorize
9y^2 + 12y + 4
On splitting middle terms we get
=9y^2+6y+6y+4
=3y(3y+2)+2(3y+2)
=(3y+2)(3y+2)

Now put this value in original polynomial
9y^2 + 12y + 4 - x^2
=(3y+2)^2-x^2

We know that (a^2-b^2) =(a+b)(a-b)
Comparing above relation with our polynomial we can write
(3y+2)^2-x^2 ={(3y+2)+x}*{(3y+2)-x}=(3y+2+x)(3y+2-x)

Solve each equation

Problem: n^2 + n = 72

Solution:
n^2+n-72=0
Split middle terms in such as a way that their product is -72n^2 and sum is +n
n^2+9n-8n-72=0
n(n+9)-8(n+9)=0
(n+9)(n-8)=0
meaning (n+9)=0 or (n-8)=0 or both
n+9=0 i.e. ...

Solution Summary

The solution describes the steps in factorizing the given polynomials. It also shows step by step method to convert given word problem into algebraic form and then finding the solutions.

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