High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Factor completely.

[tex]\[ 7x^5 - 21x^4 + 14x^3 \][/tex]

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].