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Algebra II - Factoring Polynomials

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    Study Guide Algebra II – Factoring Polynomials 1. Greatest Common Factor What Does It Mean to Factor a Polynomial? Factoring a polynomial means rewriting the polynomial as a product (a multiplication) of simpler expressions such as monomials or smaller polynomials . Because polynomials can appear in many different forms, there are several ways to factor them . One of the simplest and most common methods is factoring out the Greatest Common Factor (GCF) . What is the Greatest Common Factor? The Greatest Common Factor (GCF) is the largest factor that every term in the polynomial has in common . When we factor out the GCF, we: 1. Find the largest number or variable that divides all the terms . 2. Factor it out (place it outside parentheses). 3. Write what remains inside the parentheses. Think of it like pulling out the common part shared by every term. Example 1: Factor: Step 1: Find the common factor Both terms 5x and 5y contain 5 . So, the GCF is 5 . Step 2: Factor it out

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    Study Guide Why does this work? Divide each term by 5: • (5x ÷ 5 = x) • (5y ÷ 5 = y) So the expression becomes: Example 2: Factor: Step 1: Look for common factors First check the numbers : • 24, 16, and 8 all share factors 2, 4, and 8 • The greatest common factor is 8 Next check the variables : • Each term contains x So the GCF is (8x) . Step 2: Factor out (8x) Step 3: Check by dividing each term • (24x 3 ÷ 8x = 3x 2 ) • ( - 16x 2 ÷ 8x = - 2x) • (8x ÷ 8x = 1) So the factorization is correct:

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    Study Guide Summary • Factoring means rewriting a polynomial as a product of simpler expressions . • The Greatest Common Factor (GCF) is the largest factor shared by all terms . • To factor using the GCF: 1. Find the greatest common number and variable in all terms. 2. Factor it out. 3. Write the remaining expression inside parentheses. Example Results • (5x + 5y = 5(x + y)) • (24x 3 - 16x 2 + 8x = 8x(3x 2 - 2x + 1)) Key Idea: Always check both numbers and variables when finding the GCF. 2. Quiz: Greatest Common Factor 1. Question Factor the expression: 9x³ + 27x² − 18x Answer Choices • 9x(x² + 3x − 2) • 9x²(x + 3) − 2 • 9(x³ + 3x² − 2x) Correct Answer 9x(x² + 3x − 2)

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    Study Guide Why This Is Correct First find the Greatest Common Factor (GCF) of all terms. Terms: 9x³, 27x², − 18x Common factors: • Numbers: GCF of 9, 27, 18 = 9 • Variables: smallest power of x = x So the GCF is 9x . Factor it out: 9x³ + 27x² − 18x = 9x(x² + 3x − 2) 2. Question Factor the expression: 16x³y − 32x²y² + 24xy³ − 16xy Answer Choices • 8x(2x²y − 4xy² + 3y³ − 2y) • 8y(2x³ − 4x²y + 3xy² − 2x) • 8xy(2x² − 4xy + 3y² − 2) Correct Answer 8xy(2x² − 4xy + 3y² − 2) Why This Is Correct Find the GCF . Numbers: GCF of 16, 32, 24, 16 = 8

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    Study Guide Variables: • smallest x = x¹ • smallest y = y¹ So the GCF is 8xy . Factor it out: 16x³y − 32x²y² + 24xy³ − 16xy = 8xy(2x² − 4xy + 3y² − 2) 3. Question Factor the expression: 7x ⁵ − 28x ⁴ + 35x³ − 14x² Answer Choices  7x(x ⁴ − 4x³ + 5x² − 14x)  7x²(x³ − 4x² + 5x − 2)  7x³(x² − 4x + 5) − 14 Correct Answer 7x²(x³ − 4x² + 5x − 2) Why This Is Correct Find the GCF . Numbers: GCF of 7, 28, 35, 14 = 7 Variables: Smallest power of x = x² So the GCF is 7x² .

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    Study Guide Factor: 7x ⁵ − 28x ⁴ + 35x³ − 14x² = 7x²(x³ − 4x² + 5x − 2) 4. Question Factor the expression: 12a ⁶ + 24a ⁵ b² − 18a ⁴ b³ + 36a³b ⁴ − 12a²b ⁵ Answer Choices • 12a ⁶ + 6a²b²(4a³ − 3a²b + 6ab² − 2b³) • 6a(2a ⁵ + 4a ⁴ b² − 3a³b³ + 6a²b ⁴ − 2ab ⁵ ) • 6a²(2a ⁴ + 4a³b² − 3a²b³ + 6ab ⁴ − 2b ⁵ ) Correct Answer 6a²(2a ⁴ + 4a³b² − 3a²b³ + 6ab ⁴ − 2b ⁵ ) Why This Is Correct Find the GCF . Numbers: GCF of 12, 24, 18, 36, 12 = 6 Variables: Smallest power of a = a² So the GCF is 6a² . Factor it out: 12a ⁶ + 24a ⁵ b² − 18a ⁴ b³ + 36a³b ⁴ − 12a²b ⁵ = 6a²(2a ⁴ + 4a³b² − 3a²b³ + 6ab ⁴ − 2b ⁵ )

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    Study Guide 3. Di ff erence of Squares 1. What is the Difference of Squares? In algebra, there is a special pattern called the difference of squares . It happens when one square is subtracted from another square . A key rule to remember is: The expressions (a + b) and (a − b) are called conjugates . When you multiply conjugates, the middle terms cancel, leaving a difference of squares . 2. Example 1: Factor (x 2 - 16) Notice that both terms are perfect squares . • (x 2 = (x) 2 ) • (16 = (4) 2 ) So we can rewrite it as: Now apply the difference of squares rule: So the factored form is: 3. Example 2: Factor (25x 2 y 2 - 36z 2 ) A gain, check if both terms are perfect squares. • (25x 2 y 2 = (5xy) 2 ) • (36z 2 = (6z) 2 )

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    Study Guide R ewrite: Apply the difference of squares: 4. Example 3: Factor ((a + b) 2 - (c - d) 2 ) B oth expressions are squares. Apply the formula: Simplify inside the parentheses: 5. Example 4: Why (y 2 + 9) Cannot Be Factored Although both terms are squares: • (y 2 = (y) 2 ) • (9 = (3) 2 ) T he expression is: This is a sum of squares , not a difference of squares . Since the formula requires subtraction, this expression cannot be factored using this method .

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    Study Guide 6. Using Multiple Factoring Methods Sometimes you need to use more than one factoring method . A common strategy is: 1. First: Factor out the Greatest Common Factor (GCF) 2. Second: Check if the remaining expression is a difference of squares 7. Example 5: Factor (9x 2 - 36) First factor out the GCF (9) : Now factor the difference of squares inside the parentheses: Final answer: 8. Example 6: Factor (8(x + y) 2 - 18) First factor out the GCF (2) : Notice that: • (4(x + y) 2 = [2(x + y)] 2 ) • (9 = 3 2 ) N ow apply the difference of squares: Simplify:

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    Study Guide Summary Difference of Squares Rule Key Points to Remember • It only works when two squares are subtracted . • Both terms must be perfect squares . • The result is a pair of conjugate binomials . • A sum of squares (like (x 2 + 9)) cannot be factored using this method. • Sometimes you must factor out the GCF first , then apply the difference of squares. Steps to Factor Using Difference of Squares 1. Check if both terms are perfect squares . 2. Make sure they are being subtracted . 3. Use the formula: ((a + b)(a - b) 4. Quiz: Di ff erence of Squares 1. Question Factor the expression: 9a² − 49 Answer Choices • (3a − 7)(3a − 7) • (3a + 7)(3a − 7) • (3a + 7)(3a + 7)

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