Let’s solve the given polynomials step-by-step to find their factors.
### (1) Factor the polynomial [tex]\( x^2 - 4x - 5 \)[/tex]
1. Identify the polynomial to factor:
[tex]\[
x^2 - 4x - 5
\][/tex]
2. Find two numbers that multiply to the constant term (-5) and add to the coefficient of [tex]\( x \)[/tex] (-4):
- The numbers that satisfy this are -5 and 1 because:
[tex]\[
-5 \times 1 = -5 \quad \text{and} \quad -5 + 1 = -4
\][/tex]
3. Rewrite the middle term (-4x) using the two numbers found:
[tex]\[
x^2 - 5x + 1x - 5
\][/tex]
4. Group the terms in pairs and factor out the common factor from each pair:
[tex]\[
x(x - 5) + 1(x - 5)
\][/tex]
5. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[
(x - 5)(x + 1)
\][/tex]
So, the factors of [tex]\( x^2 - 4x - 5 \)[/tex] are:
[tex]\[
(x - 5)(x + 1)
\][/tex]
### (3) Factor the polynomial [tex]\( a^2 - 25 \)[/tex]
1. Identify the polynomial to factor:
[tex]\[
a^2 - 25
\][/tex]
2. Recognize this as a difference of squares:
[tex]\[
a^2 - 5^2
\][/tex]
3. Apply the difference of squares formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
4. Substitute [tex]\( a = a \)[/tex] and [tex]\( b = 5 \)[/tex]:
[tex]\[
a^2 - 25 = (a - 5)(a + 5)
\][/tex]
So, the factors of [tex]\( a^2 - 25 \)[/tex] are:
[tex]\[
(a - 5)(a + 5)
\][/tex]
### Summary:
- For [tex]\( x^2 - 4x - 5 \)[/tex], the factors are:
[tex]\[
(x - 5)(x + 1)
\][/tex]
- For [tex]\( a^2 - 25 \)[/tex], the factors are:
[tex]\[
(a - 5)(a + 5)
\][/tex]
