Answer :
To factor the expression [tex]\(140x^5 - 60x^4 - 35x^3 + 15x^2\)[/tex] completely, let's follow these steps:
1. Look for Common Factors:
Begin by identifying the greatest common factor (GCF) of all the terms. Here, each term contains [tex]\(x^2\)[/tex], and the coefficients 140, 60, 35, and 15 have a common factor of 5. Thus, the GCF is [tex]\(5x^2\)[/tex].
Factor out [tex]\(5x^2\)[/tex]:
[tex]\[
140x^5 - 60x^4 - 35x^3 + 15x^2 = 5x^2(28x^3 - 12x^2 - 7x + 3).
\][/tex]
2. Factor the Remaining Polynomial:
Now, we need to factor the cubic polynomial [tex]\(28x^3 - 12x^2 - 7x + 3\)[/tex].
3. Use Factoring Techniques:
We'll factor [tex]\(28x^3 - 12x^2 - 7x + 3\)[/tex] into the product of simpler polynomials.
Upon factoring, it can be rewritten as:
[tex]\[
(2x - 1)(2x + 1)(7x - 3).
\][/tex]
4. Combine Everything:
Now, combine all the factors together. The fully factored form is:
[tex]\[
5x^2(2x - 1)(2x + 1)(7x - 3).
\][/tex]
Here's a summary of the factors:
- [tex]\(2x - 1\)[/tex]
- [tex]\(2x + 1\)[/tex]
- [tex]\(7x - 3\)[/tex]
- [tex]\(5x^2\)[/tex]
These match up with some of the factors given in the list. So, the correct factors are [tex]\( (2x - 1) \)[/tex], [tex]\( (2x + 1) \)[/tex], [tex]\( (7x - 3) \)[/tex], and [tex]\( 5x^2 \)[/tex].
1. Look for Common Factors:
Begin by identifying the greatest common factor (GCF) of all the terms. Here, each term contains [tex]\(x^2\)[/tex], and the coefficients 140, 60, 35, and 15 have a common factor of 5. Thus, the GCF is [tex]\(5x^2\)[/tex].
Factor out [tex]\(5x^2\)[/tex]:
[tex]\[
140x^5 - 60x^4 - 35x^3 + 15x^2 = 5x^2(28x^3 - 12x^2 - 7x + 3).
\][/tex]
2. Factor the Remaining Polynomial:
Now, we need to factor the cubic polynomial [tex]\(28x^3 - 12x^2 - 7x + 3\)[/tex].
3. Use Factoring Techniques:
We'll factor [tex]\(28x^3 - 12x^2 - 7x + 3\)[/tex] into the product of simpler polynomials.
Upon factoring, it can be rewritten as:
[tex]\[
(2x - 1)(2x + 1)(7x - 3).
\][/tex]
4. Combine Everything:
Now, combine all the factors together. The fully factored form is:
[tex]\[
5x^2(2x - 1)(2x + 1)(7x - 3).
\][/tex]
Here's a summary of the factors:
- [tex]\(2x - 1\)[/tex]
- [tex]\(2x + 1\)[/tex]
- [tex]\(7x - 3\)[/tex]
- [tex]\(5x^2\)[/tex]
These match up with some of the factors given in the list. So, the correct factors are [tex]\( (2x - 1) \)[/tex], [tex]\( (2x + 1) \)[/tex], [tex]\( (7x - 3) \)[/tex], and [tex]\( 5x^2 \)[/tex].
