Easy Trinomial Factoring: $x^2+13x+42$

Hey there, math enthusiasts! Today, we're diving into the exciting world of algebra to tackle a common task: factoring a trinomial. Specifically, we'll be working with the expression $x^2+13x+42$. Factoring trinomials might seem a bit daunting at first, but with a little practice and understanding of the underlying principles, you'll be able to break them down with confidence. Think of it like solving a puzzle where you're trying to find two binomials that, when multiplied together, give you the original trinomial. This skill is fundamental in algebra and opens doors to solving more complex equations, simplifying expressions, and understanding quadratic functions. So, grab your thinking caps, and let's embark on this algebraic adventure to factor the trinomial $x^2+13x+42$. We'll explore the systematic approach to finding the correct factors, ensuring you not only get the right answer but also grasp why it's the right answer. Get ready to master this essential algebraic technique!

Understanding Trinomials and Factoring

Before we jump into factoring our specific trinomial, $x^2+13x+42$, let's briefly touch upon what a trinomial is and what it means to factor it. A trinomial is a polynomial that consists of three terms. Our example, $x^2+13x+42$, fits this definition perfectly, with the terms being $x^2$, $13x$, and $42$. The highest power of the variable ($x$ in this case) is 2, making it a quadratic trinomial. Factoring a trinomial, in this context, means rewriting it as a product of two binomials. A binomial is an algebraic expression with two terms. So, our goal is to find two binomials, say $(x+a)$ and $(x+b)$, such that when you multiply them, you get $x^2+13x+42$. This process is essentially the reverse of the FOIL (First, Outer, Inner, Last) method used to multiply binomials. When we multiply $(x+a)(x+b)$, we get $x^2 + bx + ax + ab$, which simplifies to $x^2 + (a+b)x + ab$. Comparing this to our target trinomial, $x^2+13x+42$, we can see a clear relationship: we need to find two numbers, $a$ and $b$, such that their sum ($a+b$) equals the coefficient of the $x$ term (which is 13), and their product ($ab$) equals the constant term (which is 42). This is the core principle we'll use to factor the trinomial $x^2+13x+42$. It's a systematic search for these two special numbers that unlock the factored form of the expression. This method is incredibly powerful because it simplifies complex algebraic expressions into more manageable components, which is crucial for further problem-solving in mathematics.

The Strategy for Factoring $x^2+13x+42$

Now, let's put our understanding into practice to factor the trinomial $x^2+13x+42$. As we established, we are looking for two numbers that add up to 13 and multiply to 42. This is where our systematic approach comes into play. We need to list pairs of numbers that multiply to 42 and then check if their sum is 13. Since our trinomial has a positive constant term (42) and a positive coefficient for the $x$ term (13), we know that both numbers we are looking for must be positive. Let's start listing the factor pairs of 42:

• 1 and 42: Their sum is $1 + 42 = 43$. This is not 13.
• 2 and 21: Their sum is $2 + 21 = 23$. This is not 13.
• 3 and 14: Their sum is $3 + 14 = 17$. This is not 13.
• 6 and 7: Their sum is $6 + 7 = 13$. Bingo! We've found our numbers.

So, the two numbers we've been searching for are 6 and 7. This means that we can factor the trinomial $x^2+13x+42$ into $(x+6)(x+7)$. To be absolutely sure, we can always check our work by multiplying these two binomials back together:

$(x+6)(x+7) = x(x+7) + 6(x+7)$ $= x^2 + 7x + 6x + 42$ $= x^2 + (7+6)x + 42$ $= x^2 + 13x + 42$

This matches our original trinomial, confirming that our factorization is correct. This step-by-step process, focusing on the sum and product of the factors, is a reliable method for factoring trinomials of this form. It's a fundamental technique that builds confidence and accuracy in algebraic manipulation. Remember, the key is to be organized and systematic in your search for the correct pair of numbers.

Evaluating the Options

We've successfully factored the trinomial $x^2+13x+42$ and found that the correct factorization is $(x+6)(x+7)$. Now, let's look at the provided options to see which one matches our result. This step is crucial to ensure we select the correct answer from the given choices:

• A. $(x+6)(x+7)$: This option exactly matches our calculated factorization. The numbers 6 and 7 add up to 13 and multiply to 42. When expanded, $(x+6)(x+7)$ indeed equals $x^2+13x+42$. This appears to be our correct answer.

• B. $(x-6)(x-7)$: Let's quickly check this. If we multiply these, we get $(x-6)(x-7) = x^2 - 7x - 6x + 42 = x^2 - 13x + 42$. The coefficient of the $x$ term is -13, not +13, so this is incorrect.

• C. $(x+3)(x+14)$: Let's examine this pair. The numbers 3 and 14 add up to $3+14=17$, not 13. While they do multiply to 42 ($3 imes 14 = 42$), their sum is incorrect for our trinomial. So, this option is also incorrect.

• D. $(x-3)(x-14)$: Similar to option B, the signs are wrong. Multiplying these gives $(x-3)(x-14) = x^2 - 14x - 3x + 42 = x^2 - 17x + 42$. The coefficient of the $x$ term is -17, which is incorrect.

By systematically evaluating each option against the requirements of our trinomial (sum of factors equals 13, product of factors equals 42), we can confidently identify the correct answer. Our initial factorization process led us directly to option A, and checking the other options confirms that A is indeed the unique correct choice for factoring the trinomial $x^2+13x+42$. This thorough evaluation ensures accuracy and reinforces the understanding of how trinomial factoring works.

Conclusion: Mastering Trinomial Factoring

We've successfully navigated the process of factoring the trinomial $x^2+13x+42$, transforming it from a sum of three terms into a product of two binomials. The key to this process lies in identifying two numbers that satisfy two conditions simultaneously: they must add up to the coefficient of the middle term (13 in this case) and multiply to the constant term (42). Through a systematic search of factor pairs of 42, we discovered that the numbers 6 and 7 perfectly met these criteria ($6+7=13$ and $6 imes 7 = 42$). This led us to the correct factored form: $(x+6)(x+7)$. We also took the crucial step of verifying our answer by multiplying the binomials back together, confirming that $(x+6)(x+7)$ indeed expands to $x^2+13x+42$. Furthermore, by examining the provided options, we were able to definitively select the correct answer, A, and understand why the other options were incorrect. Mastering trinomial factoring is a vital step in your algebraic journey, equipping you with the tools needed for solving quadratic equations, graphing parabolas, and tackling more advanced mathematical concepts. Keep practicing these techniques, and you'll find yourself becoming more proficient and confident in your algebraic abilities. Remember, the more you practice, the more intuitive this process becomes!

For further exploration into algebraic concepts and polynomial factoring, I recommend checking out resources like Khan Academy and Math is Fun.