Sari Is Factoring The Polynomial $2x^2 + 5x + 3$. If One Factor Is $(x+1)$, What Is The Other Factor?A. \$2x + 3$[/tex\] B. $3x + 2$ C. $2x - 3$ D. \$3x - 2$[/tex\]
=====================================================
Introduction
Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when we are given one of the factors. In this article, we will explore how to factor a polynomial when one of the factors is known. We will use the given polynomial $2x^2 + 5x + 3$ and the known factor $(x+1)$ to find the other factor.
Understanding the Problem
When we are given a polynomial and one of its factors, we can use the factoring method to find the other factor. The factoring method involves dividing the polynomial by the known factor to find the other factor. In this case, we are given the polynomial $2x^2 + 5x + 3$ and the known factor $(x+1)$.
Factoring the Polynomial
To factor the polynomial, we can use the division method. We will divide the polynomial $2x^2 + 5x + 3$ by the known factor $(x+1)$ to find the other factor. To do this, we can use the following steps:
- Divide the leading term of the polynomial, $2x^2$, by the leading term of the known factor, $x$.
- Multiply the known factor by the result from step 1 and subtract the product from the polynomial.
- Repeat steps 1 and 2 until the remainder is zero.
Performing the Division
Let's perform the division using the steps outlined above.
Step 1: Divide the Leading Term
The leading term of the polynomial is $2x^2$, and the leading term of the known factor is $x$. We can divide $2x^2$ by $x$ to get $2x$.
Step 2: Multiply and Subtract
We can multiply the known factor $(x+1)$ by the result from step 1, $2x$, to get $2x^2 + 2x$. We can subtract this product from the polynomial $2x^2 + 5x + 3$ to get $3x + 3$.
Step 3: Repeat the Process
We can repeat the process by dividing the leading term of the result from step 2, $3x$, by the leading term of the known factor, $x$. We can get $3$.
Step 4: Multiply and Subtract Again
We can multiply the known factor $(x+1)$ by the result from step 3, $3$, to get $3x + 3$. We can subtract this product from the result from step 2, $3x + 3$, to get $0$.
Finding the Other Factor
Since we have reached a remainder of zero, we can conclude that the other factor is $2x + 3$.
Conclusion
In this article, we have explored how to factor a polynomial when one of the factors is known. We used the given polynomial $2x^2 + 5x + 3$ and the known factor $(x+1)$ to find the other factor. We performed the division using the steps outlined above and found that the other factor is $2x + 3$.
Final Answer
The final answer is $\boxed{2x + 3}$.
=====================================================
Introduction
In our previous article, we explored how to factor a polynomial when one of the factors is known. We used the given polynomial $2x^2 + 5x + 3$ and the known factor $(x+1)$ to find the other factor. In this article, we will answer some frequently asked questions related to factoring polynomials with a given factor.
Q&A
Q: What is the first step in factoring a polynomial with a given factor?
A: The first step in factoring a polynomial with a given factor is to divide the leading term of the polynomial by the leading term of the known factor.
Q: How do I know if I have factored the polynomial correctly?
A: You can check if you have factored the polynomial correctly by multiplying the two factors together and seeing if you get the original polynomial.
Q: What if I get a remainder when I divide the polynomial by the known factor?
A: If you get a remainder when you divide the polynomial by the known factor, it means that the known factor is not a factor of the polynomial. You will need to try a different factor.
Q: Can I use the factoring method to factor a polynomial with more than two factors?
A: Yes, you can use the factoring method to factor a polynomial with more than two factors. However, you will need to use the method multiple times to factor the polynomial completely.
Q: How do I know which factor to use when factoring a polynomial?
A: When factoring a polynomial, you can use any factor that you know is a factor of the polynomial. However, it is often easiest to use a factor that is in the form of $(x-a)$, where $a$ is a constant.
Q: Can I use the factoring method to factor a polynomial with a variable in the denominator?
A: No, you cannot use the factoring method to factor a polynomial with a variable in the denominator. This is because the denominator cannot be divided by the known factor.
Examples
Example 1: Factoring a Polynomial with a Given Factor
Factor the polynomial $3x^2 + 10x + 8$ using the given factor $(x+2)$.
To factor the polynomial, we can divide the leading term of the polynomial, $3x^2$, by the leading term of the known factor, $x$. We can get $3x$. We can multiply the known factor $(x+2)$ by the result, $3x$, to get $3x^2 + 6x$. We can subtract this product from the polynomial $3x^2 + 10x + 8$ to get $4x + 8$. We can repeat the process by dividing the leading term of the result, $4x$, by the leading term of the known factor, $x$. We can get $4$. We can multiply the known factor $(x+2)$ by the result, $4$, to get $4x + 8$. We can subtract this product from the result $4x + 8$ to get $0$.
Example 2: Factoring a Polynomial with a Given Factor
Factor the polynomial $2x^2 + 7x + 3$ using the given factor $(x+3)$.
To factor the polynomial, we can divide the leading term of the polynomial, $2x^2$, by the leading term of the known factor, $x$. We can get $2x$. We can multiply the known factor $(x+3)$ by the result, $2x$, to get $2x^2 + 6x$. We can subtract this product from the polynomial $2x^2 + 7x + 3$ to get $x + 3$. We can repeat the process by dividing the leading term of the result, $x$, by the leading term of the known factor, $x$. We can get $1$. We can multiply the known factor $(x+3)$ by the result, $1$, to get $x + 3$. We can subtract this product from the result $x + 3$ to get $0$.
Conclusion
In this article, we have answered some frequently asked questions related to factoring polynomials with a given factor. We have also provided examples of how to factor polynomials using the factoring method. We hope that this article has been helpful in understanding how to factor polynomials with a given factor.
Final Answer
The final answer is that factoring a polynomial with a given factor involves dividing the polynomial by the known factor and repeating the process until the remainder is zero.