Simplify The Following Expression:B. 4 X 4 − 13 X 2 + 9 4x^4 - 13x^2 + 9 4 X 4 − 13 X 2 + 9

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting the expression in a more compact and manageable form, often by combining like terms. In this article, we will simplify the given expression: $4x^4 - 13x^2 + 9$. We will use various techniques, including factoring and substitution, to arrive at the simplified form.

Understanding the Expression

The given expression is a polynomial of degree 4, which means it has four terms. The first term is $4x^4$, the second term is $-13x^2$, and the third term is $9$. To simplify this expression, we need to identify any common factors or patterns that can be used to rewrite the expression in a more compact form.

Factoring the Expression

One way to simplify the expression is to factor out the greatest common factor (GCF) of the three terms. The GCF of $4x^4$, $-13x^2$, and $9$ is $1$, since there is no common factor other than $1$ that divides all three terms.

However, we can try to factor the expression by grouping the terms. We can group the first two terms together and factor out the common factor of $x^2$:

$$4x^4 - 13x^2 + 9 = (4x^4 - 13x^2) + 9$$

Now, we can factor out $x^2$ from the first two terms:

$$4x^4 - 13x^2 + 9 = x^2(4x^2 - 13) + 9$$

However, this does not seem to simplify the expression much. Let's try another approach.

Substitution Method

Another way to simplify the expression is to use the substitution method. We can substitute $y = x^2$ into the expression, which gives us:

$$4x^4 - 13x^2 + 9 = 4y^2 - 13y + 9$$

Now, we can try to factor the expression in terms of $y$:

$$4y^2 - 13y + 9 = (2y - 3)(2y - 3)$$

However, this is not a factorization of the original expression, since we substituted $y = x^2$. To get back to the original expression, we need to substitute $y = x^2$ back into the factorization:

$$4y^2 - 13y + 9 = (2y - 3)(2y - 3) = (2x^2 - 3)(2x^2 - 3)$$

However, this is still not the simplified form of the original expression. We need to expand the factorization to get back to the original expression:

$$(2x^2 - 3)(2x^2 - 3) = 4x^4 - 12x^2 + 9$$

This is not the original expression, since we expanded the factorization incorrectly. Let's try again.

Using the Difference of Squares Formula

We can try to simplify the expression using the difference of squares formula:

$$a^2 - b^2 = (a + b)(a - b)$$

In this case, we can rewrite the expression as:

$$4x^4 - 13x^2 + 9 = (2x^2)^2 - 13x^2 + 9$$

Now, we can apply the difference of squares formula:

$$(2x^2)^2 - 13x^2 + 9 = (2x^2 + \sqrt{13}x^2 - 3)(2x^2 - \sqrt{13}x^2 + 3)$$

However, this is not a factorization of the original expression, since we used the difference of squares formula incorrectly. The correct factorization of the original expression is:

$$(2x^2)^2 - 13x^2 + 9 = (2x^2 - 3)^2 - (4x^2 - 3)^2$$

Now, we can apply the difference of squares formula correctly:

$$(2x^2 - 3)^2 - (4x^2 - 3)^2 = (2x^2 - 3 + 4x^2 - 3)(2x^2 - 3 - 4x^2 + 3)$$

However, this is still not the simplified form of the original expression. We need to simplify the factorization further:

$$(2x^2 - 3 + 4x^2 - 3)(2x^2 - 3 - 4x^2 + 3) = (6x^2 - 6)(-2x^2)$$

Now, we can simplify the factorization further:

$$(6x^2 - 6)(-2x^2) = -12x^4 + 12x^2$$

However, this is still not the simplified form of the original expression. We need to simplify the factorization further:

$$-12x^4 + 12x^2 = -12x^2(x^2 - 1)$$

Now, we can simplify the factorization further:

$$-12x^2(x^2 - 1) = -12x^2(x - 1)(x + 1)$$

This is the simplified form of the original expression.

Conclusion

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Frequently Asked Questions

Q: What is the simplified form of the expression $4x^4 - 13x^2 + 9$?

A: The simplified form of the expression $4x^4 - 13x^2 + 9$ is $-12x^2(x - 1)(x + 1)$.

Q: How do I simplify the expression $4x^4 - 13x^2 + 9$?

A: To simplify the expression $4x^4 - 13x^2 + 9$, you can use various techniques, including factoring and substitution. You can also use the difference of squares formula to factor the expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$$a^2 - b^2 = (a + b)(a - b)$$

Q: How do I apply the difference of squares formula to the expression $4x^4 - 13x^2 + 9$?

A: To apply the difference of squares formula to the expression $4x^4 - 13x^2 + 9$, you can rewrite the expression as:

$$(2x^2)^2 - 13x^2 + 9 = (2x^2 + \sqrt{13}x^2 - 3)(2x^2 - \sqrt{13}x^2 + 3)$$

However, this is not a factorization of the original expression. The correct factorization of the original expression is:

$$(2x^2)^2 - 13x^2 + 9 = (2x^2 - 3)^2 - (4x^2 - 3)^2$$

Now, you can apply the difference of squares formula correctly:

$$(2x^2 - 3)^2 - (4x^2 - 3)^2 = (2x^2 - 3 + 4x^2 - 3)(2x^2 - 3 - 4x^2 + 3)$$

However, this is still not the simplified form of the original expression. You need to simplify the factorization further:

$$(2x^2 - 3 + 4x^2 - 3)(2x^2 - 3 - 4x^2 + 3) = (6x^2 - 6)(-2x^2)$$

Now, you can simplify the factorization further:

$$(6x^2 - 6)(-2x^2) = -12x^4 + 12x^2$$

However, this is still not the simplified form of the original expression. You need to simplify the factorization further:

$$-12x^4 + 12x^2 = -12x^2(x^2 - 1)$$

Now, you can simplify the factorization further:

$$-12x^2(x^2 - 1) = -12x^2(x - 1)(x + 1)$$

This is the simplified form of the original expression.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include:

• Factoring: This involves expressing the expression as a product of simpler expressions.
• Substitution: This involves substituting a variable or expression with a simpler one.
• Difference of squares formula: This involves expressing the expression as a difference of squares.

Q: How do I know when to use each technique?

A: You can use the following guidelines to determine which technique to use:

• If the expression can be factored, use factoring.
• If the expression can be simplified by substitution, use substitution.
• If the expression can be expressed as a difference of squares, use the difference of squares formula.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not factoring the expression correctly.
• Not substituting the variable or expression correctly.
• Not applying the difference of squares formula correctly.

Q: How do I check my work when simplifying expressions?

A: You can check your work by:

• Verifying that the simplified expression is equivalent to the original expression.
• Checking that the simplified expression is in the simplest form possible.
• Using a calculator or computer program to verify the simplified expression.

Q: What are some common applications of simplifying expressions?

A: Some common applications of simplifying expressions include:

• Solving equations and inequalities.
• Finding the roots of a polynomial.
• Simplifying complex expressions.

Q: How do I apply simplifying expressions to real-world problems?

A: You can apply simplifying expressions to real-world problems by:

• Using the simplified expression to solve a problem.
• Using the simplified expression to make a decision.
• Using the simplified expression to make a prediction.

By following these guidelines and avoiding common mistakes, you can simplify expressions effectively and apply them to real-world problems.