Answer :
To factor the expression
[tex]$$28x^3 - 7x,$$[/tex]
follow these steps:
1. Factor out the greatest common factor (GCF):
Both terms, [tex]$28x^3$[/tex] and [tex]$-7x$[/tex], have a common factor of [tex]$7x$[/tex]. Factor this out:
[tex]$$
28x^3 - 7x = 7x\left(\frac{28x^3}{7x} - \frac{7x}{7x}\right) = 7x(4x^2 - 1).
$$[/tex]
2. Factor the difference of squares:
Notice that [tex]$4x^2 - 1$[/tex] is a difference of squares. Recall that a difference of squares factors as:
[tex]$$
a^2 - b^2 = (a - b)(a + b).
$$[/tex]
Here, [tex]$4x^2$[/tex] can be written as [tex]$(2x)^2$[/tex] and [tex]$1$[/tex] as [tex]$1^2$[/tex]. Thus:
[tex]$$
4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1).
$$[/tex]
3. Write the final factored form:
Substitute the factorization from step 2 into the expression from step 1:
[tex]$$
28x^3 - 7x = 7x(2x - 1)(2x + 1).
$$[/tex]
Hence, the fully factored form of the expression is:
[tex]$$
7x(2x - 1)(2x + 1).
$$[/tex]
[tex]$$28x^3 - 7x,$$[/tex]
follow these steps:
1. Factor out the greatest common factor (GCF):
Both terms, [tex]$28x^3$[/tex] and [tex]$-7x$[/tex], have a common factor of [tex]$7x$[/tex]. Factor this out:
[tex]$$
28x^3 - 7x = 7x\left(\frac{28x^3}{7x} - \frac{7x}{7x}\right) = 7x(4x^2 - 1).
$$[/tex]
2. Factor the difference of squares:
Notice that [tex]$4x^2 - 1$[/tex] is a difference of squares. Recall that a difference of squares factors as:
[tex]$$
a^2 - b^2 = (a - b)(a + b).
$$[/tex]
Here, [tex]$4x^2$[/tex] can be written as [tex]$(2x)^2$[/tex] and [tex]$1$[/tex] as [tex]$1^2$[/tex]. Thus:
[tex]$$
4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1).
$$[/tex]
3. Write the final factored form:
Substitute the factorization from step 2 into the expression from step 1:
[tex]$$
28x^3 - 7x = 7x(2x - 1)(2x + 1).
$$[/tex]
Hence, the fully factored form of the expression is:
[tex]$$
7x(2x - 1)(2x + 1).
$$[/tex]
