Answer :
We start with the polynomial
[tex]$$2x^4 - 24x^2 + 70.$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that every term is divisible by [tex]$2$[/tex]. Factoring [tex]$2$[/tex] out gives
[tex]$$2\left(x^4 - 12x^2 + 35\right).$$[/tex]
Step 2. Recognize a quadratic in disguise.
Observe that [tex]$x^4 - 12x^2 + 35$[/tex] can be seen as a quadratic in [tex]$x^2$[/tex]. Let
[tex]$$y = x^2.$$[/tex]
Then the expression becomes
[tex]$$y^2 - 12y + 35.$$[/tex]
Step 3. Factor the quadratic in [tex]$y$[/tex].
We look for two numbers that add to [tex]$12$[/tex] and multiply to [tex]$35$[/tex]. These numbers are [tex]$5$[/tex] and [tex]$7$[/tex], because
[tex]$$5 + 7 = 12 \quad \text{and} \quad 5 \cdot 7 = 35.$$[/tex]
Thus, we can factor the quadratic as
[tex]$$y^2 - 12y + 35 = (y - 5)(y - 7).$$[/tex]
Step 4. Substitute back [tex]$x^2$[/tex].
Replacing [tex]$y$[/tex] with [tex]$x^2$[/tex], we have
[tex]$$(x^2 - 5)(x^2 - 7).$$[/tex]
Step 5. Write the complete factorization.
Substituting back into the factored form, the full factorization of the given polynomial is
[tex]$$2(x^2 - 5)(x^2 - 7).$$[/tex]
Thus, the correct choice is:
A.
[tex]$$2x^4 - 24x^2 + 70 = 2(x^2 - 5)(x^2 - 7).$$[/tex]
[tex]$$2x^4 - 24x^2 + 70.$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that every term is divisible by [tex]$2$[/tex]. Factoring [tex]$2$[/tex] out gives
[tex]$$2\left(x^4 - 12x^2 + 35\right).$$[/tex]
Step 2. Recognize a quadratic in disguise.
Observe that [tex]$x^4 - 12x^2 + 35$[/tex] can be seen as a quadratic in [tex]$x^2$[/tex]. Let
[tex]$$y = x^2.$$[/tex]
Then the expression becomes
[tex]$$y^2 - 12y + 35.$$[/tex]
Step 3. Factor the quadratic in [tex]$y$[/tex].
We look for two numbers that add to [tex]$12$[/tex] and multiply to [tex]$35$[/tex]. These numbers are [tex]$5$[/tex] and [tex]$7$[/tex], because
[tex]$$5 + 7 = 12 \quad \text{and} \quad 5 \cdot 7 = 35.$$[/tex]
Thus, we can factor the quadratic as
[tex]$$y^2 - 12y + 35 = (y - 5)(y - 7).$$[/tex]
Step 4. Substitute back [tex]$x^2$[/tex].
Replacing [tex]$y$[/tex] with [tex]$x^2$[/tex], we have
[tex]$$(x^2 - 5)(x^2 - 7).$$[/tex]
Step 5. Write the complete factorization.
Substituting back into the factored form, the full factorization of the given polynomial is
[tex]$$2(x^2 - 5)(x^2 - 7).$$[/tex]
Thus, the correct choice is:
A.
[tex]$$2x^4 - 24x^2 + 70 = 2(x^2 - 5)(x^2 - 7).$$[/tex]
