Completely Factor 3r^2 + 12r - 96

Understanding the Basics of Factoring

When we talk about factoring an expression completely, we're essentially breaking it down into its simplest multiplicative components, much like finding the prime factors of a number. For a quadratic expression like $3r^2 + 12r - 96$, this means finding the smaller expressions that, when multiplied together, result in the original expression. This process is fundamental in algebra and is crucial for solving equations, simplifying fractions, and understanding more complex mathematical concepts. The goal is to reach a point where no further common factors can be extracted from the terms. We'll approach this problem step-by-step, ensuring we don't miss any opportunities to simplify. Remember, factoring isn't just about finding a way to break down an expression; it's about finding the most complete way, which involves identifying all common factors, including numerical ones and those within the variable terms. This methodical approach ensures accuracy and efficiency. So, let's get started on dissecting this quadratic expression and revealing its fundamental building blocks.

Step 1: Identify the Greatest Common Factor (GCF)

The very first step in factoring any polynomial is to look for a greatest common factor (GCF) among all the terms. This is the largest number or expression that divides evenly into each term. In our expression, $3r^2 + 12r - 96$, let's examine the coefficients: 3, 12, and -96. We need to find the largest integer that divides all three of these numbers. The number 3 is a clear candidate, as it divides 3 (obviously), 12 (since $12 = 3 \times 4$), and -96 (since $-96 = 3 \times -32$). Since 3 is the smallest coefficient (in absolute value) and it divides the others, it is the greatest common numerical factor. There are no common variable factors across all terms because the last term, -96, does not have an 'r' term. Therefore, our GCF is simply 3. Factoring out the GCF is like 'pulling it out' of the expression. To do this, we divide each term in the original expression by the GCF (which is 3) and write the GCF on the outside of the parentheses. So, $3r^2$ divided by 3 is $r^2$, $12r$ divided by 3 is $4r$, and $-96$ divided by 3 is $-32$. This gives us the expression $3(r^2 + 4r - 32)$. This initial step is crucial because it simplifies the remaining expression, making the subsequent factoring steps much more manageable. Always be on the lookout for that GCF; it's your best friend in factoring!

Step 2: Factor the Remaining Quadratic Trinomial

Now that we've successfully factored out the GCF of 3, we are left with the quadratic trinomial inside the parentheses: $r^2 + 4r - 32$. Our next task is to factor this trinomial completely. A quadratic trinomial of the form $ax^2 + bx + c$ (where $a=1$ in our case, since the coefficient of $r^2$ is 1) can often be factored into the form $(x+p)(x+q)$. To find the values of $p$ and $q$, we need to look for two numbers that satisfy two conditions: their product must equal the constant term ($c$, which is -32 in our case), and their sum must equal the coefficient of the middle term ($b$, which is 4 in our case). So, we are searching for two numbers that multiply to -32 and add up to 4. Let's list the pairs of factors for -32 and check their sums:

• 1 and -32: Sum = -31
• -1 and 32: Sum = 31
• 2 and -16: Sum = -14
• -2 and 16: Sum = 14
• 4 and -8: Sum = -4
• -4 and 8: Sum = 4

We found our pair! The numbers -4 and 8 satisfy both conditions: their product is $(-4) \times 8 = -32$, and their sum is $-4 + 8 = 4$. Therefore, we can factor the trinomial $r^2 + 4r - 32$ into $(r - 4)(r + 8)$. This means that the expression $3(r^2 + 4r - 32)$ can be rewritten as $3(r - 4)(r + 8)$. This is the complete factorization of the original expression, as each factor $(r-4)$ and $(r+8)$ is a linear expression and cannot be factored further.

Step 3: Verify Your Factored Expression

To ensure that our factoring is correct, it's always a good practice to verify your factored expression by multiplying the factors back together. This should give us the original expression, $3r^2 + 12r - 96$. We have factored the expression into $3(r - 4)(r + 8)$. Let's start by multiplying the two binomials: $(r - 4)(r + 8)$. Using the FOIL method (First, Outer, Inner, Last):

• First: $r \times r = r^2$
• Outer: $r \times 8 = 8r$
• Inner: $-4 \times r = -4r$
• Last: $-4 \times 8 = -32$

Combining these terms, we get $r^2 + 8r - 4r - 32$, which simplifies to $r^2 + 4r - 32$. This matches the trinomial we had after factoring out the GCF, which is a good sign. Now, we need to multiply this result by the GCF we factored out earlier, which is 3:

$3 \times (r^2 + 4r - 32)$

Distribute the 3 to each term inside the parentheses:

• $3 \times r^2 = 3r^2$
• $3 \times 4r = 12r$
• $3 \times -32 = -96$

Putting it all together, we get $3r^2 + 12r - 96$. This is exactly our original expression! This verification confirms that our complete factorization of $3r^2 + 12r - 96$ into $3(r - 4)(r + 8)$ is indeed correct. This process of checking our work is a vital part of mathematical problem-solving, helping to build confidence and catch any potential errors.

Conclusion: The Complete Factorization

In summary, to factor the expression $3r^2 + 12r - 96$ completely, we followed a systematic approach. First, we identified and factored out the greatest common factor (GCF) from all terms, which was 3, leaving us with $3(r^2 + 4r - 32)$. Next, we focused on factoring the remaining quadratic trinomial, $r^2 + 4r - 32$. By finding two numbers that multiply to -32 and add to 4, we determined these numbers to be -4 and 8, allowing us to factor the trinomial into $(r - 4)(r + 8)$. Finally, we combined these steps to arrive at the complete factorization: $3(r - 4)(r + 8)$. We then verified our answer by multiplying the factors back together, confirming that it indeed equals the original expression. This step-by-step process ensures accuracy and completeness in factoring. Understanding factoring is key to many areas of mathematics, including solving quadratic equations and simplifying algebraic fractions. For further exploration into quadratic expressions and factoring techniques, you can visit Khan Academy's Algebra section, a fantastic resource for learning and practicing math concepts.