Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

It looks like you've uploaded an image of a worksheet titled "Factoring Polynomials Notes & Worksheets" for High School Algebra 1, which includes several quadratic expressions to factor completely. While the full worksheet isn't visible, I can see four problems:

1. $ 4x^2 + 4x - 35 $
2. $ 5x^2 + 37x - 24 $
7. $ 20x^2 + 2x - 6 $
8. $ 3x^2 + 25x + 42 $

Let’s solve each of these step by step using factoring techniques (like the AC method or trial and error), assuming we need to factor them completely.

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🔹 Problem 1: $ 4x^2 + 4x - 35 $

We look for two numbers that:
- Multiply to $ 4 \times (-35) = -140 $
- Add to $ 4 $

Find factors of -140 that add to 4:
- Try: $ 14 $ and $ -10 $ → $ 14 \times (-10) = -140 $, $ 14 + (-10) = 4 $ ✔

Now split the middle term:
$$
4x^2 + 14x - 10x - 35
$$
Group:
$$
(4x^2 + 14x) - (10x + 35)
= 2x(2x + 7) - 5(2x + 7)
$$
Factor out common binomial:
$$
(2x - 5)(2x + 7)
$$

✔ Answer: $ (2x - 5)(2x + 7) $

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🔹 Problem 2: $ 5x^2 + 37x - 24 $

Multiply $ 5 \times (-24) = -120 $

Find two numbers that multiply to -120 and add to 37:
- Try: $ 40 $ and $ -3 $ → $ 40 \times (-3) = -120 $, $ 40 + (-3) = 37 $ ✔

Split the middle term:
$$
5x^2 + 40x - 3x - 24
$$
Group:
$$
(5x^2 + 40x) - (3x + 24)
= 5x(x + 8) - 3(x + 8)
$$
Factor:
$$
(5x - 3)(x + 8)
$$

✔ Answer: $ (5x - 3)(x + 8) $

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🔹 Problem 7: $ 20x^2 + 2x - 6 $

First, check for a GCF:
All terms divisible by 2:
$$
2(10x^2 + x - 3)
$$

Now factor $ 10x^2 + x - 3 $

Multiply $ 10 \times (-3) = -30 $

Find two numbers that multiply to -30 and add to 1:
- Try: $ 6 $ and $ -5 $ → $ 6 \times (-5) = -30 $, $ 6 + (-5) = 1 $ ✔

Split:
$$
10x^2 + 6x - 5x - 3
= 2x(5x + 3) -1(5x + 3)
= (2x - 1)(5x + 3)
$$

So original expression:
$$
2(2x - 1)(5x + 3)
$$

✔ Answer: $ 2(2x - 1)(5x + 3) $

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🔹 Problem 8: $ 3x^2 + 25x + 42 $

Multiply $ 3 \times 42 = 126 $

Find two numbers that multiply to 126 and add to 25:
- Try: $ 18 $ and $ 7 $ → $ 18 \times 7 = 126 $, $ 18 + 7 = 25 $ ✔

Split:
$$
3x^2 + 18x + 7x + 42
= 3x(x + 6) + 7(x + 6)
= (3x + 7)(x + 6)
$$

✔ Answer: $ (3x + 7)(x + 6) $

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✔ Final Answers:

1. $ 4x^2 + 4x - 35 = (2x - 5)(2x + 7) $
2. $ 5x^2 + 37x - 24 = (5x - 3)(x + 8) $
7. $ 20x^2 + 2x - 6 = 2(2x - 1)(5x + 3) $
8. $ 3x^2 + 25x + 42 = (3x + 7)(x + 6) $

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🔍 Explanation Summary:

To factor a quadratic $ ax^2 + bx + c $:
1. Check for GCF first.
2. Use the AC Method: Multiply $ a \cdot c $, find two numbers that multiply to $ ac $ and add to $ b $.
3. Split the middle term using those numbers.
4. Group and factor by grouping.
5. Factor out the common binomial.

This method works well for trinomials where $ a \neq 1 $. Practice helps build intuition!

Let me know if you'd like a visual explanation or more problems!