Answer :
To find the Highest Common Factor (HCF) of the two given polynomials [tex]\( x^5 + 2x^4 + x^3 \)[/tex] and [tex]\( x^7 - x^5 \)[/tex], we need to determine the largest polynomial that divides both of them without leaving a remainder. Let's proceed step-by-step.
### Step 1: Factorize Each Polynomial
#### Polynomial 1: [tex]\( x^5 + 2x^4 + x^3 \)[/tex]
First, let's factor out the common factor [tex]\( x^3 \)[/tex]:
[tex]\[ x^5 + 2x^4 + x^3 = x^3(x^2 + 2x + 1) \][/tex]
We can further factorize the quadratic [tex]\( x^2 + 2x + 1 \)[/tex]:
[tex]\[ x^2 + 2x + 1 = (x + 1)^2 \][/tex]
So, the factorized form of the first polynomial is:
[tex]\[ x^5 + 2x^4 + x^3 = x^3(x + 1)^2 \][/tex]
#### Polynomial 2: [tex]\( x^7 - x^5 \)[/tex]
Factor out the common factor [tex]\( x^5 \)[/tex]:
[tex]\[ x^7 - x^5 = x^5(x^2 - 1) \][/tex]
We can further factorize the difference of squares [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ x^2 - 1 = (x + 1)(x - 1) \][/tex]
So, the factorized form of the second polynomial is:
[tex]\[ x^7 - x^5 = x^5(x + 1)(x - 1) \][/tex]
### Step 2: Determine the Common Factors
Now, we compare the factorized forms to find the common factors:
- For [tex]\( x^5 + 2x^4 + x^3 \)[/tex], we have [tex]\( x^3(x + 1)^2 \)[/tex].
- For [tex]\( x^7 - x^5 \)[/tex], we have [tex]\( x^5(x + 1)(x - 1) \)[/tex].
The common factors in both polynomials are [tex]\( x^3 \)[/tex] and [tex]\( (x + 1) \)[/tex]. However, [tex]\( x^5 \)[/tex] has [tex]\( x^3 \)[/tex] as a common factor instead of [tex]\( x^5 \)[/tex].
### Step 3: Identify the Greatest Common Factor (GCF)
Among the common factors, the least powers should be considered:
- The highest power of [tex]\( x \)[/tex] common to both polynomials is [tex]\( x^3 \)[/tex].
- The factor [tex]\( (x + 1) \)[/tex] is present in both as well, but only [tex]\( (x + 1)^1 \)[/tex] is the maximum shared across both in the smallest power form.
So, the HCF (Greatest Common Factor) of the given polynomials is:
[tex]\[ \boxed{x^3} \][/tex]
Thus, the correct answer is:
C) [tex]\( x^3 \)[/tex]
### Step 1: Factorize Each Polynomial
#### Polynomial 1: [tex]\( x^5 + 2x^4 + x^3 \)[/tex]
First, let's factor out the common factor [tex]\( x^3 \)[/tex]:
[tex]\[ x^5 + 2x^4 + x^3 = x^3(x^2 + 2x + 1) \][/tex]
We can further factorize the quadratic [tex]\( x^2 + 2x + 1 \)[/tex]:
[tex]\[ x^2 + 2x + 1 = (x + 1)^2 \][/tex]
So, the factorized form of the first polynomial is:
[tex]\[ x^5 + 2x^4 + x^3 = x^3(x + 1)^2 \][/tex]
#### Polynomial 2: [tex]\( x^7 - x^5 \)[/tex]
Factor out the common factor [tex]\( x^5 \)[/tex]:
[tex]\[ x^7 - x^5 = x^5(x^2 - 1) \][/tex]
We can further factorize the difference of squares [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ x^2 - 1 = (x + 1)(x - 1) \][/tex]
So, the factorized form of the second polynomial is:
[tex]\[ x^7 - x^5 = x^5(x + 1)(x - 1) \][/tex]
### Step 2: Determine the Common Factors
Now, we compare the factorized forms to find the common factors:
- For [tex]\( x^5 + 2x^4 + x^3 \)[/tex], we have [tex]\( x^3(x + 1)^2 \)[/tex].
- For [tex]\( x^7 - x^5 \)[/tex], we have [tex]\( x^5(x + 1)(x - 1) \)[/tex].
The common factors in both polynomials are [tex]\( x^3 \)[/tex] and [tex]\( (x + 1) \)[/tex]. However, [tex]\( x^5 \)[/tex] has [tex]\( x^3 \)[/tex] as a common factor instead of [tex]\( x^5 \)[/tex].
### Step 3: Identify the Greatest Common Factor (GCF)
Among the common factors, the least powers should be considered:
- The highest power of [tex]\( x \)[/tex] common to both polynomials is [tex]\( x^3 \)[/tex].
- The factor [tex]\( (x + 1) \)[/tex] is present in both as well, but only [tex]\( (x + 1)^1 \)[/tex] is the maximum shared across both in the smallest power form.
So, the HCF (Greatest Common Factor) of the given polynomials is:
[tex]\[ \boxed{x^3} \][/tex]
Thus, the correct answer is:
C) [tex]\( x^3 \)[/tex]
