Thank you for visiting Which polynomial can be factored using the binomial theorem A tex 25x 2 75x 225 tex B tex 25x 2 300x 225 tex C tex. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
We start with the polynomial
[tex]$$
625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625.
$$[/tex]
Step 1. Factor Out the Common Factor
Notice that each term is divisible by [tex]$625$[/tex]. Factoring [tex]$625$[/tex] out, we get
[tex]$$
625 \left( x^4 + 12x^3 + 54x^2 + 108x + 81 \right).
$$[/tex]
Step 2. Recognize the Binomial Expansion
Recall the binomial expansion formula
[tex]$$
(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
$$[/tex]
If we let [tex]$a = x$[/tex] and [tex]$b = 3$[/tex], then the expansion becomes
[tex]$$
(x+3)^4 = x^4 + 4x^3(3) + 6x^2(3^2) + 4x(3^3) + 3^4.
$$[/tex]
Calculating each term:
- [tex]$4x^3(3) = 12x^3$[/tex],
- [tex]$6x^2(9) = 54x^2$[/tex],
- [tex]$4x(27) = 108x$[/tex],
- [tex]$3^4 = 81$[/tex].
Thus,
[tex]$$
(x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81.
$$[/tex]
Step 3. Write the Final Factored Form
Substitute this result back into the factored polynomial:
[tex]$$
625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625 = 625 (x+3)^4.
$$[/tex]
Since the polynomial factors exactly as [tex]$625 (x+3)^4$[/tex], it can be factored using the binomial theorem.
Final Answer: Option 4 is the polynomial that can be factored using the binomial theorem.
[tex]$$
625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625.
$$[/tex]
Step 1. Factor Out the Common Factor
Notice that each term is divisible by [tex]$625$[/tex]. Factoring [tex]$625$[/tex] out, we get
[tex]$$
625 \left( x^4 + 12x^3 + 54x^2 + 108x + 81 \right).
$$[/tex]
Step 2. Recognize the Binomial Expansion
Recall the binomial expansion formula
[tex]$$
(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
$$[/tex]
If we let [tex]$a = x$[/tex] and [tex]$b = 3$[/tex], then the expansion becomes
[tex]$$
(x+3)^4 = x^4 + 4x^3(3) + 6x^2(3^2) + 4x(3^3) + 3^4.
$$[/tex]
Calculating each term:
- [tex]$4x^3(3) = 12x^3$[/tex],
- [tex]$6x^2(9) = 54x^2$[/tex],
- [tex]$4x(27) = 108x$[/tex],
- [tex]$3^4 = 81$[/tex].
Thus,
[tex]$$
(x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81.
$$[/tex]
Step 3. Write the Final Factored Form
Substitute this result back into the factored polynomial:
[tex]$$
625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625 = 625 (x+3)^4.
$$[/tex]
Since the polynomial factors exactly as [tex]$625 (x+3)^4$[/tex], it can be factored using the binomial theorem.
Final Answer: Option 4 is the polynomial that can be factored using the binomial theorem.
Thank you for reading the article Which polynomial can be factored using the binomial theorem A tex 25x 2 75x 225 tex B tex 25x 2 300x 225 tex C tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
- You are operating a recreational vessel less than 39 4 feet long on federally controlled waters Which of the following is a legal sound device
- Which step should a food worker complete to prevent cross contact when preparing and serving an allergen free meal A Clean and sanitize all surfaces
- For one month Siera calculated her hometown s average high temperature in degrees Fahrenheit She wants to convert that temperature from degrees Fahrenheit to degrees
Rewritten by : Jeany
