Answer :
Sure, let's factor the expression [tex]\(7x^5 - 21x^4 + 14x^3\)[/tex] completely.
Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor of all the terms in the expression. The coefficients of the terms are 7, -21, and 14. The GCF of these numbers is 7. The terms also have a common factor of [tex]\(x^3\)[/tex] since [tex]\(x^3\)[/tex] is the lowest power of [tex]\(x\)[/tex] present in all the terms.
So, the GCF of the entire expression is [tex]\(7x^3\)[/tex].
Step 2: Factor Out the GCF
Next, we factor out [tex]\(7x^3\)[/tex] from each term in the expression:
[tex]\[ 7x^5 - 21x^4 + 14x^3 = 7x^3(x^2 - 3x + 2) \][/tex]
Step 3: Factor the Quadratic Expression
Now we need to factor the quadratic [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 x term).
The numbers -1 and -2 work because:
- [tex]\(-1 \times -2 = 2\)[/tex]
- [tex]\(-1 + (-2) = -3\)[/tex]
Using these numbers, we can factor the quadratic:
[tex]\[ x^2 - 3x + 2 = (x - 1)(x - 2) \][/tex]
Step 4: Combine the Results
Finally, we combine the factored parts:
[tex]\[ 7x^5 - 21x^4 + 14x^3 = 7x^3(x - 1)(x - 2) \][/tex]
So, the completely factored form of the expression is [tex]\(7x^3(x - 1)(x - 2)\)[/tex].
Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor of all the terms in the expression. The coefficients of the terms are 7, -21, and 14. The GCF of these numbers is 7. The terms also have a common factor of [tex]\(x^3\)[/tex] since [tex]\(x^3\)[/tex] is the lowest power of [tex]\(x\)[/tex] present in all the terms.
So, the GCF of the entire expression is [tex]\(7x^3\)[/tex].
Step 2: Factor Out the GCF
Next, we factor out [tex]\(7x^3\)[/tex] from each term in the expression:
[tex]\[ 7x^5 - 21x^4 + 14x^3 = 7x^3(x^2 - 3x + 2) \][/tex]
Step 3: Factor the Quadratic Expression
Now we need to factor the quadratic [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 x term).
The numbers -1 and -2 work because:
- [tex]\(-1 \times -2 = 2\)[/tex]
- [tex]\(-1 + (-2) = -3\)[/tex]
Using these numbers, we can factor the quadratic:
[tex]\[ x^2 - 3x + 2 = (x - 1)(x - 2) \][/tex]
Step 4: Combine the Results
Finally, we combine the factored parts:
[tex]\[ 7x^5 - 21x^4 + 14x^3 = 7x^3(x - 1)(x - 2) \][/tex]
So, the completely factored form of the expression is [tex]\(7x^3(x - 1)(x - 2)\)[/tex].
