Answer :
- Find the greatest common factor (GCF) of the coefficients, which is 10.
- Identify the smallest exponent of the variable x, which is 2.
- Factor out $10x^2$ from the expression: $80x^5 - 70x^2 - 60x^7 = 10x^2(-6x^5 + 8x^3 - 7)$.
- The factored form is $\boxed{10x^2(-6x^5 + 8x^3 - 7)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the polynomial $80x^5 - 70x^2 - 60x^7$. Factoring involves finding common factors in each term and extracting them to simplify the expression.
2. Finding the GCF of Coefficients
First, let's identify the greatest common factor (GCF) of the coefficients: 80, -70, and -60. The GCF is the largest number that divides all three coefficients. We found that the GCF of 80, 70, and 60 is 10.
3. Finding the Common Variable Factor
Next, we look at the variable $x$ in each term: $x^5$, $x^2$, and $x^7$. The smallest exponent of $x$ is 2, so $x^2$ is a common factor.
4. Factoring out the GCF
Now, we factor out $10x^2$ from the entire expression: $$80x^5 - 70x^2 - 60x^7 = 10x^2(8x^3 - 7 - 6x^5)$$.
5. Rearranging the terms
Rearranging the terms inside the parenthesis in descending order of the exponent of x, we get: $$10x^2(-6x^5 + 8x^3 - 7)$$.
6. Final Answer
Thus, the factored form of the given expression is $10x^2(-6x^5 + 8x^3 - 7)$.
### Examples
Factoring polynomials is useful in many areas of mathematics and engineering. For example, in structural engineering, factoring can help simplify equations that describe the forces acting on a bridge, making it easier to analyze the bridge's stability. Similarly, in economics, factoring can be used to analyze supply and demand curves to find equilibrium points more efficiently.
- Identify the smallest exponent of the variable x, which is 2.
- Factor out $10x^2$ from the expression: $80x^5 - 70x^2 - 60x^7 = 10x^2(-6x^5 + 8x^3 - 7)$.
- The factored form is $\boxed{10x^2(-6x^5 + 8x^3 - 7)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the polynomial $80x^5 - 70x^2 - 60x^7$. Factoring involves finding common factors in each term and extracting them to simplify the expression.
2. Finding the GCF of Coefficients
First, let's identify the greatest common factor (GCF) of the coefficients: 80, -70, and -60. The GCF is the largest number that divides all three coefficients. We found that the GCF of 80, 70, and 60 is 10.
3. Finding the Common Variable Factor
Next, we look at the variable $x$ in each term: $x^5$, $x^2$, and $x^7$. The smallest exponent of $x$ is 2, so $x^2$ is a common factor.
4. Factoring out the GCF
Now, we factor out $10x^2$ from the entire expression: $$80x^5 - 70x^2 - 60x^7 = 10x^2(8x^3 - 7 - 6x^5)$$.
5. Rearranging the terms
Rearranging the terms inside the parenthesis in descending order of the exponent of x, we get: $$10x^2(-6x^5 + 8x^3 - 7)$$.
6. Final Answer
Thus, the factored form of the given expression is $10x^2(-6x^5 + 8x^3 - 7)$.
### Examples
Factoring polynomials is useful in many areas of mathematics and engineering. For example, in structural engineering, factoring can help simplify equations that describe the forces acting on a bridge, making it easier to analyze the bridge's stability. Similarly, in economics, factoring can be used to analyze supply and demand curves to find equilibrium points more efficiently.
