The volume of a prism is obtained by multiplying one of the parallel polygons and the prism's height.
2x+3, x+4, and 2x
Practice makes perfect
We will discuss how we can obtain the dimensions of the bat house model by factoring the polynomial representing the volume. We will address each of the exercise's questions one by one.
Into How Many Factors Should You Factor the Expression?
The volume of a prism is obtained by multiplying one of the parallel polygons and the prism's height.
In this case, the base is a rectangle. Since the area of a rectangle is obtained by multiplying its length times its width, we can write the formula for the volume of this prism as shown below.
V &= Area of one parallel side * Height
V &= Length * Width * Height
In accordance to the volume formula, to find expressions representing the dimensions of the bat house model, we can factor the polynomial representing its volume in three factors.
What Is the First Step in Factoring This Expression?
The first step to factoring any polynomial is to look for the greatest common factor (GCF) between all the terms. This can help to simplify the factoring process. Let's write the terms of the given polynomial, 4x^3 +22x^2 +24x, as a product of their prime factors to find the GCF.
4x^3& = 2 * 2 * x * x * x
22x^2& = 2 * 11 * x * x
24x& = 2 * 2 * 2 * 3 * x
Therefore, the GCF is 2 * x = 2x. We can now simplify our polynomial by factoring 2x out.
4x^3 +22x^2 +24x
⇕
2x(2x^2+11x+12)
Notice that the trinomial inside the parentheses is of the form ax^2+bx+c.
ax^2+ bx+ c
2x^2+ 11x+ 12
To factor a trinomial of this form we have to look for factors of a c = 24 that add up to b= 11.
Factors of 24
Sum of Factors
24,1
25
12,2
14
6,4
10
8,3
11 âś“
Since 8 * 3 = 24 and 8+3=11, we can rewrite the trinomial by writing 11x as 3x+8x. We can then make groups of two factors with the terms and then proceed to factor by grouping.
2x^2+11x+12
⇕
(2x^2+3x)+(8x+12)
Let's find the GCF of each group.
Rewrite terms as a product of their prime factors.
GCF
2x^2 &= 2 * x * x
3x &= 3 * x
x
8x &= 2 * 2 * 2 * x
12 &= 2 * 2 * 3
2 * 2 = 4
Now we can factor out the corresponding GCF from each group.
(2x^2+3x)+(8x+12)
⇕
x (2x+3)+4 (2x+3)
As we can see, there is a common binomial factor (2x+3). We can factor this binomial out to complete the factoring process.
x(2x+3)+4(2x+3)
⇕
(2x+3)(x+4)
What Expressions Can Represent the Dimensions of the Bat House?
With the work done before, we can rewrite the original polynomial completely factored.
4x^3+22x^2+24x
⇕
2x(2x+3)(x+4)
Finally, we can identify each factor as one of the bat house model's dimensions.
