Factorize 5y² + 7y And Verify Zeros Coefficients Relationship
Hey guys! Today, let's dive into a common algebra problem: factorizing the quadratic expression 5y² + 7y and verifying the relationship between its zeros and coefficients. This is a fundamental concept in algebra, and understanding it will help you tackle more complex problems down the road. So, let's break it down step by step.
Factorizing 5y² + 7y
In this section, we will delve into the factorization of the quadratic expression 5y² + 7y. Factorization is a crucial technique in algebra that simplifies expressions and helps in solving equations. When factorizing, the primary goal is to identify common factors within the expression and rewrite it as a product of simpler terms. This process not only aids in simplifying complex expressions but also facilitates the solution of equations and the identification of key properties of the expression. The ability to factorize expressions efficiently is a foundational skill in algebra, and it opens the door to solving a wide array of problems. Now, let's focus on our specific expression and see how we can apply the principles of factorization to it.
First, take a close look at the expression 5y² + 7y. Notice that both terms, 5y² and 7y, have a common factor of 'y'. This is our starting point. Factoring out 'y' from both terms is the initial step in simplifying the expression. This approach is based on the distributive property, which in reverse, allows us to pull out common factors and rewrite the expression in a more manageable form. By identifying and extracting the common factor, we lay the groundwork for further analysis and manipulation of the expression. This step is crucial because it simplifies the expression and makes it easier to work with, setting the stage for subsequent steps in solving equations or analyzing the properties of the expression.
Next, when we factor out 'y', we get y(5y + 7). This is the factored form of the expression. We've successfully rewritten the original expression as a product of two factors: 'y' and '(5y + 7)'. This factorization is a significant step because it transforms the expression from a sum of terms into a product of factors, which is incredibly useful for various algebraic manipulations. For instance, if we were solving an equation where this expression equals zero, the factored form immediately gives us potential solutions. The factored form also provides insights into the roots or zeros of the expression, which are the values of 'y' that make the expression equal to zero. By understanding the structure of the factored form, we can easily identify these roots and gain a deeper understanding of the expression's behavior.
Therefore, the factored form of 5y² + 7y is y(5y + 7). This simple yet powerful transformation is the key to unlocking further analysis and problem-solving related to this expression. By understanding how to factor out common terms, we can simplify complex expressions, solve equations, and gain insights into the fundamental properties of algebraic expressions. This factorization skill is a cornerstone of algebra and will undoubtedly be valuable in more advanced mathematical studies.
Finding the Zeros
In this section, we're going to shift our focus to finding the zeros of the factored expression y(5y + 7). Zeros, also known as roots, are the values of the variable (in this case, 'y') that make the entire expression equal to zero. Finding these zeros is a fundamental step in understanding the behavior of the expression and its graphical representation. The zeros represent the points where the graph of the expression intersects the x-axis, and they provide crucial information about the expression's solutions and properties. By identifying the zeros, we gain a deeper insight into the expression's mathematical characteristics and its applications in various contexts.
Remember, the zeros of an expression are the values of 'y' that make the expression equal to zero. This is a key concept to keep in mind as we proceed. To find these values, we utilize a fundamental principle: if a product of factors is equal to zero, then at least one of the factors must be equal to zero. This principle, known as the Zero Product Property, is the cornerstone of solving equations in factored form. It allows us to break down a complex equation into simpler ones, making it easier to find the solutions. By applying this principle, we can systematically identify the values of 'y' that satisfy the condition of the expression being equal to zero, thus revealing the zeros of the expression.
So, we set each factor equal to zero: y = 0 and 5y + 7 = 0. This step is a direct application of the Zero Product Property. By setting each factor individually to zero, we create two separate equations that are easier to solve. Each equation represents a potential zero of the expression. The first equation, y = 0, immediately gives us one zero. The second equation, 5y + 7 = 0, requires a bit more algebraic manipulation to isolate 'y' and find the corresponding zero. This process of breaking down the problem into smaller, manageable parts is a powerful technique in mathematics, allowing us to tackle complex problems step by step.
Now, solving these equations, we find that y = 0 and y = -7/5. These are the two zeros of the expression 5y² + 7y. The first zero, y = 0, is straightforward and easily identifiable. The second zero, y = -7/5, requires solving the linear equation 5y + 7 = 0. This involves subtracting 7 from both sides and then dividing by 5 to isolate 'y'. These two zeros represent the points where the graph of the expression 5y² + 7y intersects the x-axis. They are crucial values for understanding the expression's behavior and its solutions in various mathematical contexts. By finding these zeros, we have gained a significant insight into the properties of the expression.
Verifying the Relationship Between Zeros and Coefficients
This part is super important, guys! We're going to verify the relationship between the zeros we just found and the coefficients of the original quadratic expression, 5y² + 7y. This verification is a fundamental concept in algebra that highlights the connection between the roots of a quadratic equation and its coefficients. This relationship is not just a mathematical curiosity; it provides a powerful tool for checking the accuracy of our solutions and for gaining a deeper understanding of quadratic equations. By verifying this relationship, we can confirm that our calculations are correct and that we have a solid grasp of the underlying principles of quadratic equations.
For a quadratic equation in the form ay² + by + c = 0, the sum of the zeros is -b/a, and the product of the zeros is c/a. This is a crucial formula to remember. It encapsulates the relationship between the coefficients of the quadratic equation (a, b, and c) and the sum and product of its roots (zeros). This formula is derived from the properties of quadratic equations and their roots, and it provides a direct link between the equation's coefficients and the values that make the equation equal to zero. Understanding and applying this formula is essential for verifying solutions, solving problems, and gaining a deeper appreciation for the structure of quadratic equations.
In our case, a = 5, b = 7, and c = 0. This identification of the coefficients is the first step in applying the formulas for the sum and product of zeros. The coefficient 'a' is the number multiplying the y² term, 'b' is the number multiplying the y term, and 'c' is the constant term. In our expression, 5y² + 7y, we can clearly see that a = 5 and b = 7. Since there is no constant term, c = 0. These coefficients are the key ingredients in verifying the relationship between the zeros and the coefficients. By correctly identifying these values, we can accurately calculate the expected sum and product of the zeros and compare them to the values we obtained earlier.
The sum of the zeros is 0 + (-7/5) = -7/5. We found the zeros to be 0 and -7/5. To calculate their sum, we simply add these two values together. This gives us 0 + (-7/5) = -7/5. This value represents the sum of the roots of the quadratic expression. We will compare this calculated sum with the value obtained using the formula -b/a to verify the relationship between the zeros and the coefficients. If the calculated sum matches the value obtained from the formula, it provides strong evidence that our zeros are correct and that we understand the fundamental relationship between the roots and coefficients of a quadratic equation.
According to the formula, the sum should be -b/a = -7/5. This is a direct application of the formula for the sum of zeros. We substitute the values of 'b' and 'a' that we identified earlier (b = 7 and a = 5) into the formula -b/a. This gives us -7/5, which is the theoretical sum of the zeros based on the coefficients of the quadratic expression. This value will be compared with the actual sum of the zeros that we calculated earlier to verify the relationship between the roots and coefficients. If the two values match, it confirms that our calculations are accurate and that we have a solid understanding of the connection between the zeros and coefficients of a quadratic equation.
They match! Next, the product of the zeros is 0 * (-7/5) = 0. This is a straightforward calculation of the product of the two zeros we found earlier, 0 and -7/5. Multiplying these two values together, we get 0 * (-7/5) = 0. This result represents the product of the roots of the quadratic expression. We will now compare this calculated product with the value obtained using the formula c/a to verify the relationship between the zeros and the coefficients. If the calculated product matches the value obtained from the formula, it provides further evidence that our zeros are correct and that we understand the fundamental relationship between the roots and coefficients of a quadratic equation.
The formula states that the product should be c/a = 0/5 = 0. This is a direct application of the formula for the product of zeros. We substitute the values of 'c' and 'a' that we identified earlier (c = 0 and a = 5) into the formula c/a. This gives us 0/5, which simplifies to 0. This value represents the theoretical product of the zeros based on the coefficients of the quadratic expression. This value is then compared with the actual product of the zeros that we calculated earlier to verify the relationship between the roots and coefficients. If the two values match, it provides further confirmation that our calculations are accurate and that we have a strong understanding of the connection between the zeros and coefficients of a quadratic equation.
Again, they match! This verifies the relationship. Guys, by calculating both the sum and the product of the zeros and comparing them with the values obtained from the formulas -b/a and c/a, we have successfully verified the relationship between the zeros and the coefficients of the quadratic expression 5y² + 7y. This verification process not only confirms the accuracy of our calculations but also reinforces our understanding of the fundamental connection between the roots and coefficients of quadratic equations. This is a crucial concept in algebra, and mastering it will greatly enhance your problem-solving skills in more advanced mathematical topics.
Conclusion
Alright, guys! We've successfully factorized 5y² + 7y, found its zeros, and verified the relationship between the zeros and the coefficients. Remember, these are essential skills in algebra, and mastering them will set you up for success in more advanced math courses. Keep practicing, and you'll become a pro in no time!