Answer :
We start with the polynomial
[tex]$$
28x^4 - 124x^3 + 48x^2.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Observe that each term has at least a factor of [tex]$x^2$[/tex] and that the numerical coefficients are all divisible by [tex]$4$[/tex]. Thus, the GCF is [tex]$4x^2$[/tex]. Factoring [tex]$4x^2$[/tex] out of the expression gives:
[tex]$$
28x^4 - 124x^3 + 48x^2 = 4x^2\left(7x^2 - 31x + 12\right).
$$[/tex]
Step 2. Factor the quadratic [tex]$7x^2 - 31x + 12$[/tex]:
To factor the quadratic, we need to find two numbers that multiply to [tex]$7 \times 12 = 84$[/tex] and add up to [tex]$-31$[/tex]. The numbers [tex]$-3$[/tex] and [tex]$-28$[/tex] work because:
[tex]$$
-3 \times -28 = 84 \quad \text{and} \quad -3 + (-28) = -31.
$$[/tex]
Now, rewrite the middle term [tex]$-31x$[/tex] using [tex]$-3x$[/tex] and [tex]$-28x$[/tex]:
[tex]$$
7x^2 - 31x + 12 = 7x^2 - 3x - 28x + 12.
$$[/tex]
Group the terms:
[tex]$$
(7x^2 - 3x) + (-28x + 12).
$$[/tex]
Factor out the common factors in each group:
- From the first group, factor out [tex]$x$[/tex]:
[tex]$$
7x^2 - 3x = x(7x - 3).
$$[/tex]
- From the second group, factor out [tex]$-4$[/tex]:
[tex]$$
-28x + 12 = -4(7x - 3).
$$[/tex]
Now, factor out the common binomial [tex]$(7x - 3)$[/tex]:
[tex]$$
x(7x - 3) - 4(7x - 3) = (7x - 3)(x - 4).
$$[/tex]
Step 3. Write the complete factorization:
Substitute the factored form of the quadratic back into the expression:
[tex]$$
28x^4 - 124x^3 + 48x^2 = 4x^2\left(7x^2 - 31x + 12\right) = 4x^2 (7x - 3)(x - 4).
$$[/tex]
Thus, the completely factored form of the polynomial is:
[tex]$$
4x^2 (7x - 3)(x - 4).
$$[/tex]
[tex]$$
28x^4 - 124x^3 + 48x^2.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Observe that each term has at least a factor of [tex]$x^2$[/tex] and that the numerical coefficients are all divisible by [tex]$4$[/tex]. Thus, the GCF is [tex]$4x^2$[/tex]. Factoring [tex]$4x^2$[/tex] out of the expression gives:
[tex]$$
28x^4 - 124x^3 + 48x^2 = 4x^2\left(7x^2 - 31x + 12\right).
$$[/tex]
Step 2. Factor the quadratic [tex]$7x^2 - 31x + 12$[/tex]:
To factor the quadratic, we need to find two numbers that multiply to [tex]$7 \times 12 = 84$[/tex] and add up to [tex]$-31$[/tex]. The numbers [tex]$-3$[/tex] and [tex]$-28$[/tex] work because:
[tex]$$
-3 \times -28 = 84 \quad \text{and} \quad -3 + (-28) = -31.
$$[/tex]
Now, rewrite the middle term [tex]$-31x$[/tex] using [tex]$-3x$[/tex] and [tex]$-28x$[/tex]:
[tex]$$
7x^2 - 31x + 12 = 7x^2 - 3x - 28x + 12.
$$[/tex]
Group the terms:
[tex]$$
(7x^2 - 3x) + (-28x + 12).
$$[/tex]
Factor out the common factors in each group:
- From the first group, factor out [tex]$x$[/tex]:
[tex]$$
7x^2 - 3x = x(7x - 3).
$$[/tex]
- From the second group, factor out [tex]$-4$[/tex]:
[tex]$$
-28x + 12 = -4(7x - 3).
$$[/tex]
Now, factor out the common binomial [tex]$(7x - 3)$[/tex]:
[tex]$$
x(7x - 3) - 4(7x - 3) = (7x - 3)(x - 4).
$$[/tex]
Step 3. Write the complete factorization:
Substitute the factored form of the quadratic back into the expression:
[tex]$$
28x^4 - 124x^3 + 48x^2 = 4x^2\left(7x^2 - 31x + 12\right) = 4x^2 (7x - 3)(x - 4).
$$[/tex]
Thus, the completely factored form of the polynomial is:
[tex]$$
4x^2 (7x - 3)(x - 4).
$$[/tex]
