Factor The Polynomial. 3 D T − 21 D + 35 − 5 T 3dt - 21d + 35 - 5t 3 D T − 21 D + 35 − 5 T Drag The Expressions Into The Box If They Are Part Of The Factored Form Of The Polynomial.

What is Factoring a Polynomial?

Factoring a polynomial is the process of expressing it as a product of simpler polynomials, called factors. This is an essential concept in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions. In this article, we will focus on factoring a specific polynomial: $3dt - 21d + 35 - 5t$.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the given polynomial: $3dt - 21d + 35 - 5t$. We can see that it consists of four terms: three terms involving the variable $d$ and one term involving the variable $t$. Our goal is to express this polynomial as a product of simpler polynomials.

Step 1: Grouping Terms

To factor the polynomial, we need to group the terms in a way that allows us to extract common factors. Let's group the terms as follows:

• $3dt - 21d$
• $35 - 5t$

We can see that the first group has a common factor of $3d$, while the second group has a common factor of $5$.

Step 2: Factoring Out Common Factors

Now that we have grouped the terms, we can factor out the common factors:

• $3d(t - 7)$
• $5(7 - t)$

We can see that we have successfully factored out the common factors from each group.

Step 3: Combining the Factors

The final step is to combine the factors we have obtained:

$3d(t - 7) + 5(7 - t)$

However, we can simplify this expression further by combining like terms:

$3d(t - 7) - 5(t - 7)$

Now, we can see that both terms have a common factor of $(t - 7)$.

Factored Form

After combining the factors, we obtain the factored form of the polynomial:

$(3d - 5)(t - 7)$

This is the final answer.

Conclusion

Factoring a polynomial is an essential skill in algebra, and it requires a combination of grouping terms, factoring out common factors, and combining the factors. By following these steps, we can simplify complex expressions and solve equations. In this article, we have factored the polynomial $3dt - 21d + 35 - 5t$ and obtained the factored form $(3d - 5)(t - 7)$. We hope this article has provided a clear and concise guide to factoring polynomials.

Common Mistakes to Avoid

When factoring polynomials, it's essential to avoid common mistakes. Here are a few:

• Not grouping terms correctly: Make sure to group the terms in a way that allows you to extract common factors.
• Not factoring out common factors: Make sure to factor out the common factors from each group.
• Not combining the factors correctly: Make sure to combine the factors in the correct order.

Practice Problems

To practice factoring polynomials, try the following problems:

• Factor the polynomial $2x^2 + 5x + 3$
• Factor the polynomial $x^2 - 4x + 4$
• Factor the polynomial $3x^2 - 2x - 5$

Real-World Applications

Factoring polynomials has numerous real-world applications. Here are a few:

• Simplifying complex expressions: Factoring polynomials can help simplify complex expressions and make them easier to work with.
• Solving equations: Factoring polynomials can help solve equations and find the values of variables.
• Analyzing functions: Factoring polynomials can help analyze functions and understand their behavior.

Conclusion

Q: What is factoring a polynomial?

A: Factoring a polynomial is the process of expressing it as a product of simpler polynomials, called factors. This is an essential concept in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it allows us to simplify complex expressions and solve equations. It also helps us to understand the behavior of functions and make predictions about their behavior.

Q: What are some common mistakes to avoid when factoring a polynomial?

A: Some common mistakes to avoid when factoring a polynomial include:

• Not grouping terms correctly
• Not factoring out common factors
• Not combining the factors correctly

Q: How do I factor a polynomial with multiple variables?

A: To factor a polynomial with multiple variables, you need to group the terms in a way that allows you to extract common factors. You can use the distributive property to factor out common factors from each group.

Q: Can I factor a polynomial with a negative coefficient?

A: Yes, you can factor a polynomial with a negative coefficient. To do this, you need to factor out the negative sign and then factor the remaining polynomial.

Q: How do I factor a polynomial with a zero coefficient?

A: To factor a polynomial with a zero coefficient, you need to factor out the zero and then factor the remaining polynomial.

Q: Can I factor a polynomial with a fractional coefficient?

A: Yes, you can factor a polynomial with a fractional coefficient. To do this, you need to factor out the fraction and then factor the remaining polynomial.

Q: How do I factor a polynomial with a negative exponent?

A: To factor a polynomial with a negative exponent, you need to factor out the negative sign and then factor the remaining polynomial.

Q: Can I factor a polynomial with a variable in the denominator?

A: Yes, you can factor a polynomial with a variable in the denominator. To do this, you need to factor out the variable and then factor the remaining polynomial.

Q: How do I factor a polynomial with a complex number?

A: To factor a polynomial with a complex number, you need to factor out the complex number and then factor the remaining polynomial.

Q: Can I factor a polynomial with a polynomial in the numerator?

A: Yes, you can factor a polynomial with a polynomial in the numerator. To do this, you need to factor out the polynomial and then factor the remaining polynomial.

Q: How do I factor a polynomial with a polynomial in the denominator?

A: To factor a polynomial with a polynomial in the denominator, you need to factor out the polynomial and then factor the remaining polynomial.

Q: Can I factor a polynomial with a rational expression?

A: Yes, you can factor a polynomial with a rational expression. To do this, you need to factor out the rational expression and then factor the remaining polynomial.

Q: How do I factor a polynomial with a trigonometric function?

A: To factor a polynomial with a trigonometric function, you need to factor out the trigonometric function and then factor the remaining polynomial.

Q: Can I factor a polynomial with a logarithmic function?

A: Yes, you can factor a polynomial with a logarithmic function. To do this, you need to factor out the logarithmic function and then factor the remaining polynomial.

Conclusion

In conclusion, factoring a polynomial is an essential skill in algebra that requires a combination of grouping terms, factoring out common factors, and combining the factors. By following these steps, we can simplify complex expressions and solve equations. We hope this article has provided a clear and concise guide to factoring polynomials.