Answer :
Certainly! Let's factor the polynomial [tex]\(7 x^5 - 21 x^4 + 14 x^3\)[/tex] completely, step by step.
### Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor of the terms in the polynomial.
- The coefficients are 7, -21, and 14. The GCF of these numbers is 7.
- The variable part is [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. The GCF of these variables is [tex]\(x^3\)[/tex].
Thus, the GCF of the entire polynomial is [tex]\(7x^3\)[/tex].
### Step 2: Factor out the GCF
We factor out [tex]\(7x^3\)[/tex] from each term in the polynomial:
[tex]\[ 7x^3 (x^2) - 7x^3 (3x) + 7x^3 (2) = 7x^5 - 21x^4 + 14x^3 \][/tex]
This simplifies to:
[tex]\[ 7x^3 (x^2 - 3x + 2) \][/tex]
### Step 3: Factor the quadratic expression
Next, we need to factor the quadratic expression [tex]\(x^2 - 3x + 2\)[/tex]. We look for two numbers that multiply to 2 (the constant term) and add up to -3 (the coefficient of the linear term).
The two numbers that meet these criteria are -1 and -2.
Thus, the quadratic expression can be factored as:
[tex]\[ x^2 - 3x + 2 = (x - 1)(x - 2) \][/tex]
### Step 4: Write the fully factored form
Putting it all together from the factored GCF and the factored quadratic expression, we get:
[tex]\[ 7x^3 (x - 1)(x - 2) \][/tex]
### Final Answer:
So, the polynomial [tex]\(7 x^5 - 21 x^4 + 14 x^3\)[/tex] factored completely is:
[tex]\[ 7x^3 (x - 1)(x - 2) \][/tex]
Therefore,
[tex]\[ 7 x^5 - 21 x^4 + 14 x^3 = 7x^3(x - 1)(x - 2) \][/tex]
### Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor of the terms in the polynomial.
- The coefficients are 7, -21, and 14. The GCF of these numbers is 7.
- The variable part is [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex]. The GCF of these variables is [tex]\(x^3\)[/tex].
Thus, the GCF of the entire polynomial is [tex]\(7x^3\)[/tex].
### Step 2: Factor out the GCF
We factor out [tex]\(7x^3\)[/tex] from each term in the polynomial:
[tex]\[ 7x^3 (x^2) - 7x^3 (3x) + 7x^3 (2) = 7x^5 - 21x^4 + 14x^3 \][/tex]
This simplifies to:
[tex]\[ 7x^3 (x^2 - 3x + 2) \][/tex]
### Step 3: Factor the quadratic expression
Next, we need to factor the quadratic expression [tex]\(x^2 - 3x + 2\)[/tex]. We look for two numbers that multiply to 2 (the constant term) and add up to -3 (the coefficient of the linear term).
The two numbers that meet these criteria are -1 and -2.
Thus, the quadratic expression can be factored as:
[tex]\[ x^2 - 3x + 2 = (x - 1)(x - 2) \][/tex]
### Step 4: Write the fully factored form
Putting it all together from the factored GCF and the factored quadratic expression, we get:
[tex]\[ 7x^3 (x - 1)(x - 2) \][/tex]
### Final Answer:
So, the polynomial [tex]\(7 x^5 - 21 x^4 + 14 x^3\)[/tex] factored completely is:
[tex]\[ 7x^3 (x - 1)(x - 2) \][/tex]
Therefore,
[tex]\[ 7 x^5 - 21 x^4 + 14 x^3 = 7x^3(x - 1)(x - 2) \][/tex]
