Answer :
We want to factor the polynomial
[tex]$$
x^3 - 7x^2 - 5x + 35.
$$[/tex]
A good strategy is to use factoring by grouping. Follow these steps:
1. First, rewrite the polynomial by grouping the first two terms and the last two terms:
[tex]$$
x^3 - 7x^2 - 5x + 35 = (x^3 - 7x^2) + (-5x + 35).
$$[/tex]
2. Factor out the common factors in each group:
- In the first group, [tex]$x^3 - 7x^2$[/tex], factor out [tex]$x^2$[/tex]:
[tex]$$
x^3 - 7x^2 = x^2 (x - 7).
$$[/tex]
- In the second group, [tex]$-5x + 35$[/tex], factor out [tex]$-5$[/tex]:
[tex]$$
-5x + 35 = -5 (x - 7).
$$[/tex]
3. Now the expression becomes
[tex]$$
x^2 (x-7) - 5 (x-7).
$$[/tex]
4. Notice that both terms contain the common factor [tex]$(x-7)$[/tex]. Factor out [tex]$(x-7)$[/tex]:
[tex]$$
x^2 (x-7) - 5 (x-7) = (x-7)(x^2-5).
$$[/tex]
Thus, the polynomial factors as
[tex]$$
x^3-7x^2-5x+35 = (x-7)(x^2-5).
$$[/tex]]
[tex]$$
x^3 - 7x^2 - 5x + 35.
$$[/tex]
A good strategy is to use factoring by grouping. Follow these steps:
1. First, rewrite the polynomial by grouping the first two terms and the last two terms:
[tex]$$
x^3 - 7x^2 - 5x + 35 = (x^3 - 7x^2) + (-5x + 35).
$$[/tex]
2. Factor out the common factors in each group:
- In the first group, [tex]$x^3 - 7x^2$[/tex], factor out [tex]$x^2$[/tex]:
[tex]$$
x^3 - 7x^2 = x^2 (x - 7).
$$[/tex]
- In the second group, [tex]$-5x + 35$[/tex], factor out [tex]$-5$[/tex]:
[tex]$$
-5x + 35 = -5 (x - 7).
$$[/tex]
3. Now the expression becomes
[tex]$$
x^2 (x-7) - 5 (x-7).
$$[/tex]
4. Notice that both terms contain the common factor [tex]$(x-7)$[/tex]. Factor out [tex]$(x-7)$[/tex]:
[tex]$$
x^2 (x-7) - 5 (x-7) = (x-7)(x^2-5).
$$[/tex]
Thus, the polynomial factors as
[tex]$$
x^3-7x^2-5x+35 = (x-7)(x^2-5).
$$[/tex]]
