Factor Of G^2-28g+196: Find It Now!

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Hey guys! Let's break down this math problem together. We're on a mission to figure out which of the following is a factor of the quadratic expression $g^2 - 28g + 196$. This is a classic factoring problem, and with a little bit of algebraic maneuvering, we'll nail it. So, let's dive right in and make math a little less mysterious!

Understanding the Problem

Before we jump into solving, let’s make sure we understand what the question is asking. We are given a quadratic expression, which is a polynomial of degree two. In simpler terms, it’s an expression that includes a variable raised to the power of 2 (like $g^2$), along with other terms involving the variable and constants. Our mission is to find which of the provided options (A, B, C, or D) is a factor of this expression. A factor is an expression that divides evenly into the given quadratic expression without leaving a remainder.

To find the factor, we'll need to factorize the quadratic expression $g^2 - 28g + 196$. Factoring involves breaking down the quadratic expression into a product of two binomials (expressions with two terms). Once we have these binomials, we can easily identify which of the given options matches one of our factors. This involves recognizing patterns, applying algebraic techniques, and staying sharp on our multiplication and addition rules. So, let’s get started and break this down step by step!

Method 1: Factoring the Quadratic Expression

The most straightforward approach to this problem is to factor the quadratic expression $g^2 - 28g + 196$. Factoring involves rewriting the expression as a product of two binomials. Here’s how we can do it:

Step 1: Recognize the Pattern

First, notice that the given expression might be a perfect square trinomial. A perfect square trinomial has the form $(a - b)^2 = a^2 - 2ab + b^2$ or $(a + b)^2 = a^2 + 2ab + b^2$.

In our case, we have $g^2 - 28g + 196$. We need to check if it fits the perfect square trinomial pattern. To do this, we examine the first and last terms to see if they are perfect squares. The first term, $g^2$, is indeed a perfect square, as it is $(g)^2$. The last term, 196, is also a perfect square because $14^2 = 196$. So, we can consider $g$ as $a$ and 14 as $b$.

Step 2: Verify the Middle Term

Now, let's verify if the middle term, $-28g$, fits the pattern. According to the perfect square trinomial formula, the middle term should be $-2ab$. In our case, this would be $-2 * g * 14 = -28g$, which matches the middle term in our expression. This confirms that $g^2 - 28g + 196$ is indeed a perfect square trinomial.

Step 3: Write the Factored Form

Since we've confirmed that the expression is a perfect square trinomial, we can now write it in its factored form. Using the formula $(a - b)^2 = a^2 - 2ab + b^2$, we can rewrite our expression as:

(g−14)2(g - 14)^2

This means that $g^2 - 28g + 196 = (g - 14)(g - 14)$.

Step 4: Identify the Factor

Now that we have the factored form, we can easily identify the factors of the original expression. From $(g - 14)(g - 14)$, we see that $(g - 14)$ is a factor.

Comparing this with the given options:

  • A. $(g - 7)$
  • B. $(g - 14)$
  • C. $(g + 14)$
  • D. $(g + 7)$

We can see that option B, $(g - 14)$, matches one of our factors. Therefore, $(g - 14)$ is a factor of $g^2 - 28g + 196$.

So, the correct answer is B. $g - 14$!

Method 2: Testing the Options

Another way to solve this problem is by testing each of the given options to see which one is a factor of the quadratic expression $g^2 - 28g + 196$. This method involves dividing the quadratic expression by each option and checking if the remainder is zero. If the remainder is zero, then the option is a factor.

Step 1: Test Option A: $(g - 7)$

To test if $(g - 7)$ is a factor, we can perform polynomial long division or use synthetic division. Alternatively, we can check if $g = 7$ is a root of the quadratic expression. To do this, substitute $g = 7$ into the expression:

(7)2−28(7)+196=49−196+196=49(7)^2 - 28(7) + 196 = 49 - 196 + 196 = 49

Since the result is not zero, $(g - 7)$ is not a factor.

Step 2: Test Option B: $(g - 14)$

Now, let's test if $(g - 14)$ is a factor by substituting $g = 14$ into the expression:

(14)2−28(14)+196=196−392+196=0(14)^2 - 28(14) + 196 = 196 - 392 + 196 = 0

Since the result is zero, $(g - 14)$ is a factor of the quadratic expression.

Step 3: Test Option C: $(g + 14)$

To test if $(g + 14)$ is a factor, substitute $g = -14$ into the expression:

(−14)2−28(−14)+196=196+392+196=784(-14)^2 - 28(-14) + 196 = 196 + 392 + 196 = 784

Since the result is not zero, $(g + 14)$ is not a factor.

Step 4: Test Option D: $(g + 7)$

Finally, let's test if $(g + 7)$ is a factor by substituting $g = -7$ into the expression:

(−7)2−28(−7)+196=49+196+196=441(-7)^2 - 28(-7) + 196 = 49 + 196 + 196 = 441

Since the result is not zero, $(g + 7)$ is not a factor.

Step 5: Identify the Correct Factor

After testing each option, we found that only $(g - 14)$ results in zero when substituted into the quadratic expression. Therefore, $(g - 14)$ is the correct factor.

So, using this method, we also find that the correct answer is B. $(g - 14)$.

Conclusion

Alright, guys! We've tackled this problem using two different methods: factoring the quadratic expression and testing the options. Both methods led us to the same conclusion: the correct factor of $g^2 - 28g + 196$ is $(g - 14)$. So, the answer is B. Great job, and keep up the awesome work!

Whether you prefer to factorize directly or test each option, understanding the underlying principles is key. Keep practicing, and these types of problems will become second nature! You've got this!