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Answer :
- Find the greatest common factor (GCF) of the coefficients, which is 10.
- Identify the lowest power of x, which is $x^2$.
- Factor out the GCF $10x^2$ from the expression: $10x^2(8x^3 - 7 - 6x^5)$.
- Rearrange the terms and factor out -1: $-10x^2(6x^5 - 8x^3 + 7)$.
The final factored form is $\boxed{{-10x^2(6x^5 - 8x^3 + 7)}}$.
### Explanation
1. Identifying the GCF
We are asked to factor the polynomial $80 x^5-70 x^2-60 x^7$. First, we identify the greatest common factor (GCF) of the coefficients and the variable $x$.
2. Finding the GCF of Coefficients
The coefficients are 80, -70, and -60. The GCF of these numbers is 10. We can confirm this by finding the prime factorization of each number:
$80 = 2^4 \times 5$
$70 = 2 \times 5 \times 7$
$60 = 2^2 \times 3 \times 5$
The common factors are 2 and 5, so the GCF is $2 \times 5 = 10$.
3. Finding the Lowest Power of x
The variable $x$ appears in each term with different exponents. The smallest exponent of $x$ is 2, so we can factor out $x^2$ from each term.
4. Factoring out the GCF
Therefore, the GCF of the entire expression is $10x^2$. We factor out $10x^2$ from the expression:
$80 x^5-70 x^2-60 x^7 = 10x^2(8x^3 - 7 - 6x^5)$
5. Rearranging Terms
Rearrange the terms inside the parenthesis in descending order of the exponent of $x$:
$10x^2(8x^3 - 7 - 6x^5) = 10x^2(-6x^5 + 8x^3 - 7)$
6. Factoring out -1
Factor out -1 from the parenthesis:
$10x^2(-6x^5 + 8x^3 - 7) = -10x^2(6x^5 - 8x^3 + 7)$
7. Final Factored Form
The expression inside the parenthesis, $6x^5 - 8x^3 + 7$, cannot be factored easily using simple techniques. Thus, the factored form of the given expression is $-10x^2(6x^5 - 8x^3 + 7)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In economics, factoring can help in modeling supply and demand curves. Understanding how to factor polynomials allows you to break down complex problems into simpler, more manageable parts, making it easier to find solutions and make informed decisions.
- Identify the lowest power of x, which is $x^2$.
- Factor out the GCF $10x^2$ from the expression: $10x^2(8x^3 - 7 - 6x^5)$.
- Rearrange the terms and factor out -1: $-10x^2(6x^5 - 8x^3 + 7)$.
The final factored form is $\boxed{{-10x^2(6x^5 - 8x^3 + 7)}}$.
### Explanation
1. Identifying the GCF
We are asked to factor the polynomial $80 x^5-70 x^2-60 x^7$. First, we identify the greatest common factor (GCF) of the coefficients and the variable $x$.
2. Finding the GCF of Coefficients
The coefficients are 80, -70, and -60. The GCF of these numbers is 10. We can confirm this by finding the prime factorization of each number:
$80 = 2^4 \times 5$
$70 = 2 \times 5 \times 7$
$60 = 2^2 \times 3 \times 5$
The common factors are 2 and 5, so the GCF is $2 \times 5 = 10$.
3. Finding the Lowest Power of x
The variable $x$ appears in each term with different exponents. The smallest exponent of $x$ is 2, so we can factor out $x^2$ from each term.
4. Factoring out the GCF
Therefore, the GCF of the entire expression is $10x^2$. We factor out $10x^2$ from the expression:
$80 x^5-70 x^2-60 x^7 = 10x^2(8x^3 - 7 - 6x^5)$
5. Rearranging Terms
Rearrange the terms inside the parenthesis in descending order of the exponent of $x$:
$10x^2(8x^3 - 7 - 6x^5) = 10x^2(-6x^5 + 8x^3 - 7)$
6. Factoring out -1
Factor out -1 from the parenthesis:
$10x^2(-6x^5 + 8x^3 - 7) = -10x^2(6x^5 - 8x^3 + 7)$
7. Final Factored Form
The expression inside the parenthesis, $6x^5 - 8x^3 + 7$, cannot be factored easily using simple techniques. Thus, the factored form of the given expression is $-10x^2(6x^5 - 8x^3 + 7)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In economics, factoring can help in modeling supply and demand curves. Understanding how to factor polynomials allows you to break down complex problems into simpler, more manageable parts, making it easier to find solutions and make informed decisions.
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