Sometimes, you know the product and are asked to find the factors. This process is called factoring.
For example, suppose you want to paint a rectangle on a wall
and you only have enough paint to cover 20 square feet. If the length of each
side must be an integer, what are the dimensions of all the possible rectangles
you could paint?
Recall that the formula for the area of a rectangle is A =
lw. If A = 20 square feet, then the measures of the length and width
of the painted rectangle must be factor pairs of 20. The factor pairs of 20 are
1 and 20, 2 and 10, and 4 and 5. The figures below show rectangles with these factors
as measures of length and width.
In Chapter 9, you used the Distributive Property to multiply
a polynomial by a monomial.
You can reverse this process to express a polynomial in factored
form. A polynomial is in factored form when it is expressed as the product
of polynomials. For example, to factor 8y2 + 10y, find
the greatest common factor of 8y2 and 10y.
The GCF of 8y2 and 10y is 2y.
Write each term as a product of the GCF and its remaining factors. Then use the
Distributive Property.
8y2 + 10y written in factored form
is 2y(4y + 5).
Factor each polynomial.
1.
30x2 + 12x
Alternative Solutions:
First, find the GCF of 30x2
and 12x.
The GCF of 30x2 and 12x is 6x. Write each term as a product of the GCF and its remaining factors.
2.
15ab2 – 25abc
Alternative Solutions:
The GCF is 5ab.
Example
Factor each polynomial.
3.
18x2y + 12xy2 + 6xy
Alternative Solutions:
The
GCF is 6xy. When 6xy is factored from 6xy the remaining factor
is 1.
4.
7x2 + 9yz
Alternative Solutions:
There
are no common factors of 7x2 and 9yz other than 1.
Therefore,
7x2 + 9yz cannot be factored using the GCF. It is a
prime polynomial.
If you know a product and one of its factors, you can use
division to find the other factor. To divide a polynomial by a monomial, divide
each term of the polynomial by the monomial.
Example
5.
Divide 15x3 + 12x2 by 3x.
Alternative Solutions:
Therefore,
(15x3 + 12x2) ÷ 3x = 5x2
+ 4x.
Factoring a polynomial can help simplify computations.
Landscaping
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6. A stone walkway is to be built around a square planter that contains a
shade tree.
A. If the walkway is 2 meters wide,
write an expression in factored form that represents the area of the walkway.
Alternative Solutions:
Let x
represent the length and width of the planter. You can find the area of the
walkway by finding the sum of the areas of the 8 rectangular sections shown in
the figure.
The resulting expression can be simplified by using the Distributive Property to combine like terms and then factoring.
B. If the dimensions of the square planter are 1.5 meters by 1.5 meters,
find the area of the walkway.
Alternative Solutions:
The area of the walkway is 28 square meters.
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