Simplify The Expression: $40x^3 + 56x^2 - 35x - 49$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression: $40x^3 + 56x^2 - 35x - 49$. We will use various techniques such as factoring, combining like terms, and using the distributive property to simplify the expression.

Understanding the Expression

The given expression is a polynomial expression of degree 3, which means it has three terms with different powers of x. The expression is: $40x^3 + 56x^2 - 35x - 49$. To simplify this expression, we need to understand the properties of exponents and how to combine like terms.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $x^3$ means $x \times x \times x$. When we multiply two or more numbers with the same base, we can add their exponents. For example, $x^3 \times x^2 = x^{3+2} = x^5$.

Combining Like Terms

Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable x and the exponent 2. To combine like terms, we add their coefficients. For example, $2x^2 + 3x^2 = 5x^2$.

Factoring the Expression

To simplify the expression, we can try to factor it. Factoring involves expressing an expression as a product of simpler expressions. In this case, we can try to factor the expression by grouping the terms.

Grouping Terms

We can group the terms in the expression as follows:

$40x^3 + 56x^2 - 35x - 49 = (40x^3 + 56x^2) - (35x + 49)$

Now, we can factor out the greatest common factor (GCF) from each group.

Factoring Out the GCF

The GCF of $40x^3$ and $56x^2$ is $8x^2$. The GCF of $-35x$ and $-49$ is $-7$. Therefore, we can factor out the GCF from each group as follows:

$(40x^3 + 56x^2) - (35x + 49) = 8x^2(5x + 7) - 7(5x + 7)$

Now, we can factor out the common factor $(5x + 7)$ from each group.

Factoring Out the Common Factor

We can factor out the common factor $(5x + 7)$ from each group as follows:

$8x^2(5x + 7) - 7(5x + 7) = (5x + 7)(8x^2 - 7)$

Therefore, the simplified expression is: $(5x + 7)(8x^2 - 7)$.

Conclusion

In this article, we simplified the expression $40x^3 + 56x^2 - 35x - 49$ using various techniques such as factoring, combining like terms, and using the distributive property. We factored out the greatest common factor (GCF) from each group and then factored out the common factor from each group. The simplified expression is: $(5x + 7)(8x^2 - 7)$.

Frequently Asked Questions

• Q: What is the degree of the simplified expression? A: The degree of the simplified expression is 3.
• Q: What is the greatest common factor (GCF) of the expression? A: The GCF of the expression is 1.
• Q: How do I simplify an expression using factoring? A: To simplify an expression using factoring, you need to group the terms and then factor out the greatest common factor (GCF) from each group.

Final Answer

The final answer is: $\boxed{(5x + 7)(8x^2 - 7)}$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression: $40x^3 + 56x^2 - 35x - 49$. We will use various techniques such as factoring, combining like terms, and using the distributive property to simplify the expression.

Understanding the Expression

The given expression is a polynomial expression of degree 3, which means it has three terms with different powers of x. The expression is: $40x^3 + 56x^2 - 35x - 49$. To simplify this expression, we need to understand the properties of exponents and how to combine like terms.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $x^3$ means $x \times x \times x$. When we multiply two or more numbers with the same base, we can add their exponents. For example, $x^3 \times x^2 = x^{3+2} = x^5$.

Combining Like Terms

Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable x and the exponent 2. To combine like terms, we add their coefficients. For example, $2x^2 + 3x^2 = 5x^2$.

Factoring the Expression

To simplify the expression, we can try to factor it. Factoring involves expressing an expression as a product of simpler expressions. In this case, we can try to factor the expression by grouping the terms.

Grouping Terms

We can group the terms in the expression as follows:

$40x^3 + 56x^2 - 35x - 49 = (40x^3 + 56x^2) - (35x + 49)$

Now, we can factor out the greatest common factor (GCF) from each group.

Factoring Out the GCF

The GCF of $40x^3$ and $56x^2$ is $8x^2$. The GCF of $-35x$ and $-49$ is $-7$. Therefore, we can factor out the GCF from each group as follows:

$(40x^3 + 56x^2) - (35x + 49) = 8x^2(5x + 7) - 7(5x + 7)$

Now, we can factor out the common factor $(5x + 7)$ from each group.

Factoring Out the Common Factor

We can factor out the common factor $(5x + 7)$ from each group as follows:

$8x^2(5x + 7) - 7(5x + 7) = (5x + 7)(8x^2 - 7)$

Therefore, the simplified expression is: $(5x + 7)(8x^2 - 7)$.

Q&A

Q: What is the degree of the simplified expression?

A: The degree of the simplified expression is 3.

Q: What is the greatest common factor (GCF) of the expression?

A: The GCF of the expression is 1.

Q: How do I simplify an expression using factoring?

A: To simplify an expression using factoring, you need to group the terms and then factor out the greatest common factor (GCF) from each group.

Q: What is the difference between combining like terms and factoring?

A: Combining like terms involves adding or subtracting terms with the same variable and exponent, while factoring involves expressing an expression as a product of simpler expressions.

Q: Can I simplify an expression using only combining like terms?

A: Yes, you can simplify an expression using only combining like terms. However, factoring can often be a more efficient and effective way to simplify an expression.

Q: How do I know when to use factoring and when to use combining like terms?

A: You should use factoring when the expression can be expressed as a product of simpler expressions, and you should use combining like terms when the expression can be simplified by adding or subtracting terms with the same variable and exponent.

Q: Can I simplify an expression that has no common factors?

A: Yes, you can simplify an expression that has no common factors by using combining like terms.

Q: How do I check my work when simplifying an expression?

A: You should check your work by plugging the simplified expression back into the original expression and verifying that it is true.

Conclusion

In this article, we simplified the expression $40x^3 + 56x^2 - 35x - 49$ using various techniques such as factoring, combining like terms, and using the distributive property. We also answered some frequently asked questions about simplifying expressions.

Final Answer

The final answer is: $\boxed{(5x + 7)(8x^2 - 7)}$