Simplify The Expression: $6x^2 + 48x + 72$

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms. In this article, we will simplify the expression $6x^2 + 48x + 72$ using various techniques.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$. Here, $a = 6$, $b = 48$, and $c = 72$. To simplify this expression, we need to factor out the greatest common factor (GCF) of the terms.

Factoring Out the GCF

The GCF of the terms $6x^2$, $48x$, and $72$ is $6$. We can factor out $6$ from each term:

$$6x^2 + 48x + 72 = 6(x^2 + 8x + 12)$$

Simplifying the Expression

Now, we need to simplify the expression inside the parentheses, $x^2 + 8x + 12$. We can try to factor this expression by finding two numbers whose product is $12$ and whose sum is $8$. These numbers are $4$ and $4$, since $4 \times 4 = 16$ and $4 + 4 = 8$. However, this is not the correct factorization, as the product is not $12$.

Using the Perfect Square Trinomial Formula

The expression $x^2 + 8x + 12$ can be rewritten as a perfect square trinomial using the formula $(x + a)^2 = x^2 + 2ax + a^2$. In this case, we have:

$$x^2 + 8x + 12 = (x + 4)^2 + 4$$

However, this is not the correct factorization, as the expression is not a perfect square trinomial.

Using the Difference of Squares Formula

The expression $x^2 + 8x + 12$ can be rewritten as a difference of squares using the formula $(x + a)^2 - b^2 = (x + a + b)(x + a - b)$. However, this expression does not fit this formula.

Using the Sum and Difference of Cubes Formula

The expression $x^2 + 8x + 12$ can be rewritten as a sum and difference of cubes using the formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. However, this expression does not fit this formula.

Using the Rational Root Theorem

The rational root theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. In this case, the constant term is $12$ and the leading coefficient is $1$. The factors of $12$ are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. Since the leading coefficient is $1$, the only possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Using Synthetic Division

We can use synthetic division to divide the polynomial $x^2 + 8x + 12$ by the possible rational roots. Let's try dividing by $1$:

$$\begin{array}{c|rrr} 1 & 1 & 8 & 12 \\ & & 1 & 9 \\ \hline & 1 & 9 & 21 \end{array}$$

The remainder is $21$, which is not equal to $0$. Therefore, $1$ is not a root of the polynomial.

Using the Quadratic Formula

The quadratic formula states that the roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = 8$, and $c = 12$. Plugging these values into the formula, we get:

$$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(12)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-8 \pm \sqrt{64 - 48}}{2}$$

$$x = \frac{-8 \pm \sqrt{16}}{2}$$

$$x = \frac{-8 \pm 4}{2}$$

Therefore, the roots of the polynomial are $x = -6$ and $x = -2$.

Conclusion

In this article, we simplified the expression $6x^2 + 48x + 72$ using various techniques. We factored out the GCF, tried to factor the expression inside the parentheses, and used the quadratic formula to find the roots of the polynomial. The final simplified expression is:

$$6x^2 + 48x + 72 = 6(x + 2)(x + 6)$$

This expression can be further simplified by factoring out the GCF of the terms inside the parentheses.

Final Answer

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Introduction

In our previous article, we simplified the expression $6x^2 + 48x + 72$ using various techniques. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q: What is the greatest common factor (GCF) of the terms $6x^2$, $48x$, and $72$?

A: The GCF of the terms $6x^2$, $48x$, and $72$ is $6$. We can factor out $6$ from each term to simplify the expression.

Q: How do you factor out the GCF from an expression?

A: To factor out the GCF from an expression, we need to identify the GCF of the terms and then divide each term by the GCF. In this case, we factored out $6$ from each term to get:

$$6x^2 + 48x + 72 = 6(x^2 + 8x + 12)$$

Q: What is the perfect square trinomial formula?

A: The perfect square trinomial formula is $(x + a)^2 = x^2 + 2ax + a^2$. We can use this formula to rewrite an expression as a perfect square trinomial.

Q: Can you give an example of how to use the perfect square trinomial formula?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can rewrite this expression as a perfect square trinomial using the formula:

$$x^2 + 8x + 12 = (x + 4)^2 + 4$$

However, this is not the correct factorization, as the expression is not a perfect square trinomial.

Q: What is the difference of squares formula?

A: The difference of squares formula is $(x + a)^2 - b^2 = (x + a + b)(x + a - b)$. We can use this formula to rewrite an expression as a difference of squares.

Q: Can you give an example of how to use the difference of squares formula?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can rewrite this expression as a difference of squares using the formula:

However, this expression does not fit this formula.

Q: What is the sum and difference of cubes formula?

A: The sum and difference of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. We can use this formula to rewrite an expression as a sum and difference of cubes.

Q: Can you give an example of how to use the sum and difference of cubes formula?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can rewrite this expression as a sum and difference of cubes using the formula:

However, this expression does not fit this formula.

Q: What is the rational root theorem?

A: The rational root theorem states that if a rational number $p/q$ is a root of the polynomial $f(x)$, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.

Q: Can you give an example of how to use the rational root theorem?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can use the rational root theorem to find the possible rational roots of the polynomial. The factors of the constant term $12$ are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. Since the leading coefficient is $1$, the only possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It involves using a table to divide the polynomial by the linear factor.

Q: Can you give an example of how to use synthetic division?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can use synthetic division to divide the polynomial by the possible rational roots. Let's try dividing by $1$:

$$\begin{array}{c|rrr} 1 & 1 & 8 & 12 \\ & & 1 & 9 \\ \hline & 1 & 9 & 21 \end{array}$$

The remainder is $21$, which is not equal to $0$. Therefore, $1$ is not a root of the polynomial.

Q: What is the quadratic formula?

A: The quadratic formula is a method of finding the roots of a quadratic equation. It states that the roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: Can you give an example of how to use the quadratic formula?

A: Yes, let's consider the expression $x^2 + 8x + 12$. We can use the quadratic formula to find the roots of the polynomial. Plugging in the values $a = 1$, $b = 8$, and $c = 12$, we get:

$$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(12)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-8 \pm \sqrt{64 - 48}}{2}$$

$$x = \frac{-8 \pm \sqrt{16}}{2}$$

$$x = \frac{-8 \pm 4}{2}$$

Therefore, the roots of the polynomial are $x = -6$ and $x = -2$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $6x^2 + 48x + 72$. We discussed various techniques for simplifying expressions, including factoring out the GCF, using the perfect square trinomial formula, and using the quadratic formula. We also provided examples of how to use these techniques to simplify expressions.

Final Answer

The final answer is $\boxed{6(x + 2)(x + 6)}$.