Question

factoring teams activity

1. look at this polynomial with your team. write the expression that represents the area of the polynomial in your notes. hint: it will begin

xy + x² + 6x + 9y + 18

1. recall that the word \sum\ means addition. you have just written an expression that is a sum. you will now write an expression that is a produ

what does the word product mean?

1. write an expression that represents the product (length)(width) for this rectangle. hint: the length will begin

y + x +.....
(y + x + )( )
length width

1. use the box method to multiply these two binomials in your notes. answer should be the polynomial that represents total area.

Explanation:

Step1: Factor the polynomial \(xy + x^{2}+6x + 9y+18\)

Group the terms: \((xy + x^{2})+(9y + 9x)+(6x+18)\)
Factor out common factors from each group: \(x(y + x)+9(y + x)+6(x + 3)\)
Notice that \((x + y)\) is a common factor for the first two - grouped terms: \((x + y)(x + 9)+6(x + 3)\)
Another way is to rewrite the polynomial as \(x^{2}+xy+9y + 6x+18\)
Group as \(x^{2}+(y + 6)x+(9y + 18)\)
Factor the quadratic in \(x\): \(x^{2}+(y + 6)x+9(y + 2)\)
Using the formula for factoring \(ax^{2}+bx + c\) (\(a = 1\)), we look for two numbers \(m\) and \(n\) such that \(m + n=y + 6\) and \(mn=9(y + 2)\)
We can also use the box - method (area model) for factoring.
The length of the rectangle: count the number of \(y\) - units, \(x\) - units and \(1\) - units along one side. The number of \(y\) - units is \(1\), the number of \(x\) - units is \(1\), and the number of \(1\) - units is \(6\). So the length \(l=y + x+6\)
The width: count the number of \(x\) - units and \(1\) - units along the perpendicular side. The number of \(x\) - units is \(1\) and the number of \(1\) - units is \(3\). So the width \(w=x + 3\)

Step2: Multiply the length and width \((y + x+6)(x + 3)\) using the distributive property (FOIL - like for trinomials and binomials)

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Answer:

The product (length)(width) is \((x + y+6)(x + 3)\)