Answer :
To factor the expression
[tex]$$20x - 45x^3,$$[/tex]
follow these steps:
1. Factor out the greatest common factor (GCF):
Notice that both terms have a factor of [tex]$5x$[/tex]. Factoring [tex]$5x$[/tex] out gives:
[tex]$$20x - 45x^3 = 5x\left(\frac{20x}{5x} - \frac{45x^3}{5x}\right) = 5x\left(4 - 9x^2\right).$$[/tex]
2. Factor the remaining expression:
The expression inside the parentheses, [tex]$4 - 9x^2$[/tex], is a difference of two squares because [tex]$4$[/tex] is [tex]$2^2$[/tex] and [tex]$9x^2$[/tex] is [tex]$(3x)^2$[/tex]. Recall the difference of two squares formula:
[tex]$$a^2 - b^2 = (a - b)(a + b).$$[/tex]
Here, [tex]$a = 2$[/tex] and [tex]$b = 3x$[/tex]. Applying the formula:
[tex]$$4 - 9x^2 = (2 - 3x)(2 + 3x).$$[/tex]
3. Write the final factored form:
Substitute the factorization back into the expression:
[tex]$$20x - 45x^3 = 5x (2 - 3x)(2 + 3x).$$[/tex]
It is also acceptable to write [tex]$(2 + 3x)$[/tex] as [tex]$(3x + 2)$[/tex]; therefore, the completely factored form of the expression is:
[tex]$$5x(2 - 3x)(3x + 2).$$[/tex]
This is the final, completely factored form of the given expression.
[tex]$$20x - 45x^3,$$[/tex]
follow these steps:
1. Factor out the greatest common factor (GCF):
Notice that both terms have a factor of [tex]$5x$[/tex]. Factoring [tex]$5x$[/tex] out gives:
[tex]$$20x - 45x^3 = 5x\left(\frac{20x}{5x} - \frac{45x^3}{5x}\right) = 5x\left(4 - 9x^2\right).$$[/tex]
2. Factor the remaining expression:
The expression inside the parentheses, [tex]$4 - 9x^2$[/tex], is a difference of two squares because [tex]$4$[/tex] is [tex]$2^2$[/tex] and [tex]$9x^2$[/tex] is [tex]$(3x)^2$[/tex]. Recall the difference of two squares formula:
[tex]$$a^2 - b^2 = (a - b)(a + b).$$[/tex]
Here, [tex]$a = 2$[/tex] and [tex]$b = 3x$[/tex]. Applying the formula:
[tex]$$4 - 9x^2 = (2 - 3x)(2 + 3x).$$[/tex]
3. Write the final factored form:
Substitute the factorization back into the expression:
[tex]$$20x - 45x^3 = 5x (2 - 3x)(2 + 3x).$$[/tex]
It is also acceptable to write [tex]$(2 + 3x)$[/tex] as [tex]$(3x + 2)$[/tex]; therefore, the completely factored form of the expression is:
[tex]$$5x(2 - 3x)(3x + 2).$$[/tex]
This is the final, completely factored form of the given expression.
