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Answer :
- Group the terms: $(12x^3 + 10x^2) + (30x + 25)$.
- Factor out common factors: $2x^2(6x + 5) + 5(6x + 5)$.
- Factor out the common binomial: $(6x + 5)(2x^2 + 5)$.
- The factored form is $\boxed{(6x + 5)(2x^2 + 5)}$.
### Explanation
1. Understanding the Problem
We are given the polynomial $12x^3 + 10x^2 + 30x + 25$ and asked to factor it.
2. Grouping Terms
We will attempt to factor by grouping. First, we group the first two terms and the last two terms: $(12x^3 + 10x^2) + (30x + 25)$.
3. Factoring out Common Factors
Next, we factor out the greatest common factor from each group. From the first group, $12x^3 + 10x^2$, we can factor out $2x^2$, which gives us $2x^2(6x + 5)$. From the second group, $30x + 25$, we can factor out $5$, which gives us $5(6x + 5)$. So we have $2x^2(6x + 5) + 5(6x + 5)$.
4. Factoring out the Common Binomial
Now, we factor out the common binomial factor $(6x + 5)$ from the entire expression: $(6x + 5)(2x^2 + 5)$.
5. Checking for Real Roots
The quadratic $2x^2 + 5$ has no real roots, since $2x^2 + 5 = 0$ implies $2x^2 = -5$, so $x^2 = -\frac{5}{2}$, which has no real solutions. Therefore, the factorization is $(6x + 5)(2x^2 + 5)$.
6. Final Factorization
Thus, the factored form of the polynomial $12x^3 + 10x^2 + 30x + 25$ is $(6x + 5)(2x^2 + 5)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model market behavior, and computer scientists use it to design efficient algorithms. In simple terms, imagine you are designing a rectangular garden and need to determine the dimensions based on the total area and some constraints. Factoring can help you find the possible lengths and widths that satisfy the given conditions, ensuring your garden fits perfectly in the available space.
- Factor out common factors: $2x^2(6x + 5) + 5(6x + 5)$.
- Factor out the common binomial: $(6x + 5)(2x^2 + 5)$.
- The factored form is $\boxed{(6x + 5)(2x^2 + 5)}$.
### Explanation
1. Understanding the Problem
We are given the polynomial $12x^3 + 10x^2 + 30x + 25$ and asked to factor it.
2. Grouping Terms
We will attempt to factor by grouping. First, we group the first two terms and the last two terms: $(12x^3 + 10x^2) + (30x + 25)$.
3. Factoring out Common Factors
Next, we factor out the greatest common factor from each group. From the first group, $12x^3 + 10x^2$, we can factor out $2x^2$, which gives us $2x^2(6x + 5)$. From the second group, $30x + 25$, we can factor out $5$, which gives us $5(6x + 5)$. So we have $2x^2(6x + 5) + 5(6x + 5)$.
4. Factoring out the Common Binomial
Now, we factor out the common binomial factor $(6x + 5)$ from the entire expression: $(6x + 5)(2x^2 + 5)$.
5. Checking for Real Roots
The quadratic $2x^2 + 5$ has no real roots, since $2x^2 + 5 = 0$ implies $2x^2 = -5$, so $x^2 = -\frac{5}{2}$, which has no real solutions. Therefore, the factorization is $(6x + 5)(2x^2 + 5)$.
6. Final Factorization
Thus, the factored form of the polynomial $12x^3 + 10x^2 + 30x + 25$ is $(6x + 5)(2x^2 + 5)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model market behavior, and computer scientists use it to design efficient algorithms. In simple terms, imagine you are designing a rectangular garden and need to determine the dimensions based on the total area and some constraints. Factoring can help you find the possible lengths and widths that satisfy the given conditions, ensuring your garden fits perfectly in the available space.
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