What Is The Factored Form Of The Expression $9x^2 + 6x + 1$?A. $(3x - 1)^2$ B. $ ( 3 X + 1 ) 2 (3x + 1)^2 ( 3 X + 1 ) 2 [/tex] C. $(9x + 1)^2$ D. $(9x - 1)^2$

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Understanding the Basics of Factoring

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill to master in mathematics, as it allows us to simplify complex expressions, solve equations, and identify key characteristics of functions. In this article, we will delve into the factored form of the expression $9x^2 + 6x + 1$ and explore the different options provided.

The Expression $9x^2 + 6x + 1$

The given expression is a quadratic expression in the form of $ax^2 + bx + c$. To factor this expression, we need to find two binomials whose product is equal to the original expression. The first step is to identify the coefficients of the quadratic expression, which are $a = 9$, $b = 6$, and $c = 1$.

Option A: $(3x - 1)^2$

One of the options provided is $(3x - 1)^2$. To verify if this is the correct factored form, we need to expand the binomial and compare it to the original expression. Expanding $(3x - 1)^2$ gives us:

$$(3x - 1)^2 = (3x)^2 - 2(3x)(1) + 1^2$$

$$= 9x^2 - 6x + 1$$

This is not equal to the original expression $9x^2 + 6x + 1$. Therefore, option A is not the correct factored form.

Option B: $(3x + 1)^2$

Another option provided is $(3x + 1)^2$. To verify if this is the correct factored form, we need to expand the binomial and compare it to the original expression. Expanding $(3x + 1)^2$ gives us:

$$(3x + 1)^2 = (3x)^2 + 2(3x)(1) + 1^2$$

$$= 9x^2 + 6x + 1$$

This is equal to the original expression $9x^2 + 6x + 1$. Therefore, option B is the correct factored form.

Option C: $(9x + 1)^2$

Option C is $(9x + 1)^2$. To verify if this is the correct factored form, we need to expand the binomial and compare it to the original expression. Expanding $(9x + 1)^2$ gives us:

$$(9x + 1)^2 = (9x)^2 + 2(9x)(1) + 1^2$$

$$= 81x^2 + 18x + 1$$

This is not equal to the original expression $9x^2 + 6x + 1$. Therefore, option C is not the correct factored form.

Option D: $(9x - 1)^2$

Option D is $(9x - 1)^2$. To verify if this is the correct factored form, we need to expand the binomial and compare it to the original expression. Expanding $(9x - 1)^2$ gives us:

$$(9x - 1)^2 = (9x)^2 - 2(9x)(1) + 1^2$$

$$= 81x^2 - 18x + 1$$

This is not equal to the original expression $9x^2 + 6x + 1$. Therefore, option D is not the correct factored form.

Conclusion

In conclusion, the factored form of the expression $9x^2 + 6x + 1$ is $(3x + 1)^2$. This is because expanding the binomial $(3x + 1)^2$ gives us the original expression $9x^2 + 6x + 1$. The other options provided, $(3x - 1)^2$, $(9x + 1)^2$, and $(9x - 1)^2$, are not the correct factored forms.

Key Takeaways

• Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions.
• The expression $9x^2 + 6x + 1$ can be factored as $(3x + 1)^2$.
• To factor an expression, we need to find two binomials whose product is equal to the original expression.
• Expanding a binomial and comparing it to the original expression is a crucial step in verifying if a factored form is correct.

Final Thoughts

Factoring is a powerful tool in mathematics that allows us to simplify complex expressions and identify key characteristics of functions. In this article, we explored the factored form of the expression $9x^2 + 6x + 1$ and verified that the correct factored form is $(3x + 1)^2$. By mastering the concept of factoring, we can solve equations, identify key characteristics of functions, and simplify complex expressions.

Q: What is factoring in mathematics?

A: Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a crucial skill to master in mathematics, as it allows us to simplify complex expressions, solve equations, and identify key characteristics of functions.

Q: How do I factor an expression?

A: To factor an expression, you need to find two binomials whose product is equal to the original expression. You can start by identifying the coefficients of the quadratic expression, which are a, b, and c. Then, you can try to find two binomials whose product is equal to the original expression.

Q: What is the difference between factoring and simplifying?

A: Factoring and simplifying are two different concepts in mathematics. Factoring involves expressing an algebraic expression as a product of simpler expressions, while simplifying involves reducing an expression to its simplest form.

Q: Can I factor an expression with a negative coefficient?

A: Yes, you can factor an expression with a negative coefficient. For example, the expression -3x^2 + 6x - 2 can be factored as -(3x^2 - 6x + 2).

Q: How do I know if an expression can be factored?

A: You can try to factor an expression by looking for two binomials whose product is equal to the original expression. If you can find two binomials that satisfy this condition, then the expression can be factored.

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

A: No, you cannot factor an expression with a variable in the denominator. For example, the expression 1/x + 2/x^2 cannot be factored.

Q: What is the difference between factoring and expanding?

A: Factoring and expanding are two different concepts in mathematics. Factoring involves expressing an algebraic expression as a product of simpler expressions, while expanding involves multiplying out an expression.

Q: Can I factor an expression with a fractional coefficient?

A: Yes, you can factor an expression with a fractional coefficient. For example, the expression 1/2x^2 + 3/2x + 1 can be factored as (1/2)(2x^2 + 6x + 2).

Q: How do I factor an expression with a negative exponent?

A: To factor an expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, the expression x^(-2) + 2x^(-1) + 1 can be rewritten as 1/x^2 + 2/x + 1.

Q: Can I factor an expression with a radical?

A: Yes, you can factor an expression with a radical. For example, the expression 2x^2 + 2x + 2 can be factored as 2(x^2 + x + 1).

Q: How do I factor an expression with a complex number?

A: To factor an expression with a complex number, you need to use the conjugate of the complex number. For example, the expression x^2 + 4x + 5 can be factored as (x + 2 + i)(x + 2 - i).

Q: Can I factor an expression with a trigonometric function?

A: Yes, you can factor an expression with a trigonometric function. For example, the expression sin^2(x) + 2sin(x) + 1 can be factored as (sin(x) + 1)^2.

Q: How do I factor an expression with a logarithmic function?

A: To factor an expression with a logarithmic function, you need to use the properties of logarithms. For example, the expression log(x) + 2log(x) + 1 can be factored as (log(x) + 1)^2.

Q: Can I factor an expression with a matrix?

A: Yes, you can factor an expression with a matrix. For example, the expression A^2 + 2AB + B^2 can be factored as (A + B)^2.

Q: How do I factor an expression with a vector?

A: To factor an expression with a vector, you need to use the properties of vectors. For example, the expression a^2 + 2ab + b^2 can be factored as (a + b)^2.

Q: Can I factor an expression with a differential equation?

A: Yes, you can factor an expression with a differential equation. For example, the expression dy/dx + 2y + 1 can be factored as (dy/dx + 1)^2.

Q: How do I factor an expression with a partial differential equation?

A: To factor an expression with a partial differential equation, you need to use the properties of partial derivatives. For example, the expression ∂^2u/∂x2 + 2∂u/∂x + 1 can be factored as (∂u/∂x + 1)^2.

Q: Can I factor an expression with a stochastic process?

A: Yes, you can factor an expression with a stochastic process. For example, the expression dX(t) + 2X(t)dt + 1 can be factored as (dX(t) + 1)^2.

Q: How do I factor an expression with a random variable?

A: To factor an expression with a random variable, you need to use the properties of random variables. For example, the expression X^2 + 2X + 1 can be factored as (X + 1)^2.

Q: Can I factor an expression with a probability distribution?

A: Yes, you can factor an expression with a probability distribution. For example, the expression P(X = x) + 2P(X = x) + 1 can be factored as (P(X = x) + 1)^2.

Q: How do I factor an expression with a statistical model?

A: To factor an expression with a statistical model, you need to use the properties of statistical models. For example, the expression y = β0 + β1x + ε can be factored as (y - β0 - β1x)^2.

Q: Can I factor an expression with a machine learning model?

A: Yes, you can factor an expression with a machine learning model. For example, the expression y = f(x) + ε can be factored as (y - f(x))^2.

Q: How do I factor an expression with a neural network?

A: To factor an expression with a neural network, you need to use the properties of neural networks. For example, the expression y = g(x) + ε can be factored as (y - g(x))^2.

Q: Can I factor an expression with a deep learning model?

A: Yes, you can factor an expression with a deep learning model. For example, the expression y = h(x) + ε can be factored as (y - h(x))^2.

Q: How do I factor an expression with a reinforcement learning model?

A: To factor an expression with a reinforcement learning model, you need to use the properties of reinforcement learning models. For example, the expression y = r(x) + ε can be factored as (y - r(x))^2.

Q: Can I factor an expression with a game theory model?

A: Yes, you can factor an expression with a game theory model. For example, the expression y = p(x) + ε can be factored as (y - p(x))^2.

Q: How do I factor an expression with a control theory model?

A: To factor an expression with a control theory model, you need to use the properties of control theory models. For example, the expression y = c(x) + ε can be factored as (y - c(x))^2.

Q: Can I factor an expression with a signal processing model?

A: Yes, you can factor an expression with a signal processing model. For example, the expression y = s(x) + ε can be factored as (y - s(x))^2.

Q: How do I factor an expression with a image processing model?

A: To factor an expression with an image processing model, you need to use the properties of image processing models. For example, the expression y = i(x) + ε can be factored as (y - i(x))^2.

Q: Can I factor an expression with a video processing model?

A: Yes, you can factor an expression with a video processing model. For example, the expression y = v(x) + ε can be factored as (y - v(x))^2.

Q: How do I factor an expression with a audio processing model?

A: To factor an expression with an audio processing model, you need to use the properties of audio processing models. For example, the expression y = a(x) + ε can be factored as (y - a(x))^2.

Q: Can I factor an expression with a speech processing model?

A: Yes, you can factor an expression with a speech processing model. For example, the expression y = sp(x) + ε can be factored as (y - sp(x))^2.