Rivalry2

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Original 2026-01-01

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1 In this section, we will focus on three main methods used to solve polynomials.

2 Greatest Common Factor

3 In this method, we take out the largest<a>number</a>(and variable, if any) that all the terms have in common. It’s like reversing the<a>distributive property</a>.

4 Distributive property:

5 p(q + r) = pq + pr

6 Factored form:

7 pq + pr = p(q + r), where p is the greatest<a>common factor</a>.

8 Example: Factor the polynomial 12x + 18

9 Step 1: Find the GCF

10 Break each term into its factors:

11 \(12x = 3 × 4 × x\) or \(6 × 2 × x\)

12 \(18 = 3 × 6\)

13 So, the GCF is 6.

14 Step 2:Factor it out

15 \(12x + 18 = 6(2x + 3)\)

16 Step 3:Check by distributing

17 \(6(2x + 3) = 6 \times 2x + 6 \times 3 = 12x + 18\)

18 Factoring polynomials by grouping

19 This method works well while trying to factor<a>trinomials</a>(three-term expressions) that can’t be factored just by taking out a common factor. The idea is to split the middle term into two parts so we can group the terms and factor in pairs.

20 Example: Factor \(x^2 + 11x + 24 = (x + 3)(x + 8)\)

21 Step 1:Find two numbers

22 Look for two numbers that:

23 Add up to 11 (the middle term), and

24 Multiply by 24 (the last term)

25 The numbers 3 and 8 work because:

26 \(3 + 8 = 11\)

27 \(3 × 8 = 24\)

28 Step 2: Rewrite the middle term using these two numbers

29 \(x^2 + 11x + 24 = x^2 + 3x + 8x + 24\)

30 Step 3:group the terms and factor

31 \((x^2 + 3x) + (8x + 24)\)

32 \(x(x + 3) + 8(x + 3) \)

33 Take the common<a>binomial</a>factor,

34 \((x + 3)(x + 8)\)

35 So, \((x + 3)(x + 8)\) are the factors of \(x^2 + 11x + 24\)

36 Factoring using Identities

37 Algebraic identities can also help in factorizing polynomials. Commonly used identities for this purpose are:

38 \((a + b)^2 = a^2 + 2ab + b^2\)

39 \((a - b)^2 = a^2 - 2ab + b^2\)

40 \(a^2 - b^2 = (a + b)(a - b)\)

41 Let’s take examples for each identity listed above.

42 Example 1:Factorize \(x^2 + 6x + 9\), using (a + b)2

43 Step 1 - Compare with the identity

44 \(x^2 + 6x + 9 = x^2 + 2 \times 3 \times x + 3^2\)

45 We can see the pattern a2 + 2ab + b2, where a = x and b = 3

46 Step 2 - Apply the identity

47 \(x^2 + 6x + 9 = (x + 3)^2\)

48 So, (x + 3)2 is the answer.

49 Example 2:Factor \(x^2 - 10x + 25\) using identity (a - b)2

50 Step 1 - Match with the identity

51 \(x^2 - 10x + 25 = x^2 - 2 \times 5 \times x + 5^2\)

52 Here, a = x and b = 5

53 Step 2 - Apply the identity

54 \(x^2 - 10x + 25 = (x - 5)^2\)

55 (x - 5)2 is the answer.

56 Example 3:Factor x2 - 49 using identity a2 - b2

57 Step 1 - It needs to be the difference between two<a>perfect squares</a>

58 \(x^2 - 49 = x^2 - 7^2\)

59 Step 2 - Apply Identity a2 - b2 = (a + b)(a - b)

60 So, \(x^2 - 49 = (x + 7)(x - 7)\)