Simplify And Factor Expressions: A Step-by-Step Guide

by ADMIN 54 views

Hey guys! Let's dive into simplifying and factoring some expressions. This is a fundamental concept in algebra, and mastering it will definitely help you in more advanced topics. We'll go through each problem step by step, so you can see exactly how it's done. Let's get started!

17. Simplifying and Factoring 4(10k + 1) - 4(k - 4)

Okay, so our first expression is 4(10k + 1) - 4(k - 4). The goal here is to first simplify this expression by distributing and combining like terms, and then factor it if possible. This process involves a couple of key steps that, once mastered, become second nature. Let's break it down:

  1. Distribution: The first thing we need to do is distribute the numbers outside the parentheses to the terms inside. So, we'll distribute the first 4 across (10k + 1) and the second -4 across (k - 4). Remember, it's super important to pay attention to the signs!

    • 4 * 10k = 40k
    • 4 * 1 = 4
    • -4 * k = -4k
    • -4 * -4 = 16 (Remember, a negative times a negative is a positive!)

  2. Rewriting the Expression: Now, let's rewrite the entire expression with the distributed terms. We get: 40k + 4 - 4k + 16

  3. Combining Like Terms: Next, we need to combine the terms that are alike. In this case, we have 40k and -4k as well as 4 and 16. Let's combine them:

    • 40k - 4k = 36k
    • 4 + 16 = 20

  4. Simplified Expression: After combining like terms, our expression looks much simpler: 36k + 20

  5. Factoring: Now, let's see if we can factor this expression. Factoring involves finding the greatest common factor (GCF) of the terms and pulling it out. The GCF of 36k and 20 is 4. So, we can factor out a 4:

    • 36k ÷ 4 = 9k
    • 20 ÷ 4 = 5

  6. Final Factored Form: So, we rewrite the expression by factoring out the 4: 4(9k + 5)

Therefore, the simplified and factored form of 4(10k + 1) - 4(k - 4) is 4(9k + 5). See? Not so scary when we break it down step by step!

18. Simplifying and Factoring -3(w + 2) - 12(2 - w)

Next up, we have the expression -3(w + 2) - 12(2 - w). Just like before, we'll tackle this by distributing, combining like terms, and then factoring. Remember, the key is to take it one step at a time. Let’s jump in!

  1. Distribution: First, let's distribute the -3 across (w + 2) and the -12 across (2 - w). Again, keep a close eye on those signs – they can be tricky!

    • -3 * w = -3w
    • -3 * 2 = -6
    • -12 * 2 = -24
    • -12 * -w = 12w (A negative times a negative gives us a positive!)

  2. Rewriting the Expression: Now, let’s put the distributed terms back into the expression: -3w - 6 - 24 + 12w

  3. Combining Like Terms: Now, let's combine the like terms. We have -3w and 12w, and we also have -6 and -24. Let's combine them:

    • -3w + 12w = 9w
    • -6 - 24 = -30

  4. Simplified Expression: After combining like terms, our expression becomes: 9w - 30

  5. Factoring: Now, let’s see if we can factor anything out. We need to find the greatest common factor (GCF) of 9w and -30. The GCF here is 3.

    • 9w ÷ 3 = 3w
    • -30 ÷ 3 = -10

  6. Final Factored Form: Let's factor out the 3: 3(3w - 10)

So, the simplified and factored form of -3(w + 2) - 12(2 - w) is 3(3w - 10). We're on a roll!

19. Simplifying and Factoring 4(r - 9s) + 6(r + 2s)

Moving right along, we have 4(r - 9s) + 6(r + 2s). By now, you're probably getting the hang of this. Distribute, combine, and factor – that's the name of the game. Let’s do it!

  1. Distribution: First up, let's distribute the 4 across (r - 9s) and the 6 across (r + 2s):

    • 4 * r = 4r
    • 4 * -9s = -36s
    • 6 * r = 6r
    • 6 * 2s = 12s

  2. Rewriting the Expression: Let's rewrite the expression with the distributed terms: 4r - 36s + 6r + 12s

  3. Combining Like Terms: Now, we combine the like terms. We have 4r and 6r, and we also have -36s and 12s. Let's combine them:

    • 4r + 6r = 10r
    • -36s + 12s = -24s

  4. Simplified Expression: After combining like terms, the expression becomes: 10r - 24s

  5. Factoring: Time to factor! We need to find the greatest common factor (GCF) of 10r and -24s. The GCF here is 2.

    • 10r ÷ 2 = 5r
    • -24s ÷ 2 = -12s

  6. Final Factored Form: Let's factor out the 2: 2(5r - 12s)

So, the simplified and factored form of 4(r - 9s) + 6(r + 2s) is 2(5r - 12s). Awesome!

20. Simplifying and Factoring 9(x + 3y) - (x - y)

Last but not least, we have 9(x + 3y) - (x - y). This one has a little twist because we're subtracting the entire second expression, but we've got this! Same steps apply: distribute, combine, and factor. Here we go!

  1. Distribution: Let's distribute the 9 across (x + 3y). For the second part, remember that subtracting (x - y) is the same as distributing a -1 across it:

    • 9 * x = 9x
    • 9 * 3y = 27y
    • -1 * x = -x
    • -1 * -y = y (Again, negative times negative is positive!)

  2. Rewriting the Expression: Now, let’s rewrite the expression with the distributed terms: 9x + 27y - x + y

  3. Combining Like Terms: Combine the like terms. We have 9x and -x, and we also have 27y and y. Let's combine them:

    • 9x - x = 8x
    • 27y + y = 28y

  4. Simplified Expression: After combining like terms, our expression becomes: 8x + 28y

  5. Factoring: Time to factor! We need to find the greatest common factor (GCF) of 8x and 28y. The GCF here is 4.

    • 8x ÷ 4 = 2x
    • 28y ÷ 4 = 7y

  6. Final Factored Form: Let's factor out the 4: 4(2x + 7y)

So, the simplified and factored form of 9(x + 3y) - (x - y) is 4(2x + 7y). You nailed it!

Conclusion

Alright, we've tackled four different expressions, simplifying and factoring each one step by step. Remember, the key to mastering these problems is to practice and take it one step at a time. Distribute carefully, combine like terms accurately, and find the greatest common factor for factoring. You've got this! Keep practicing, and you'll become a factoring pro in no time. Keep up the great work, guys!