Unit 5: Polynomial Functions - Factoring Polynomials Study Guide

Fundamental Rules of Factoring Polynomials\n\n- Difference of Squares: The rule states that for any binomial involving the difference of two squared terms, the factors are the sum and difference of the square roots of those terms. The formula is: \n $a^2 - b^2 = (a - b)(a + b)$\n\n- Sum of Cubes: This identity is used to factor a binomial where two cubed terms are added. The formula is: \n $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$\n\n- Difference of Cubes: This identity is used to factor a binomial where one cubed term is subtracted from another. The formula is: \n $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$\n\n# General Instructions for Factoring Polynomials\n\n- Check for GCF First: Before applying any specific factoring rule or identity, you must always look for a Greatest Common Factor (GCF) that can be factored out of all terms in the polynomial. This simplifies the remaining expression and is essential for complete factoring.\n- Complete Factoring: The directions specify that every polynomial must be factored completely. This means that if a factor can be factored further (e.g., a resulting quadratic binomial is a difference of squares), you must continue the process until no further factoring is possible.\n- Unit Information: This material is part of Unit 5: Polynomial Functions from Homework 2: Factoring Polynomials.\n- Copyright: This educational content is provided by \u00a9 Gina Wilson (All Things Algebra\u00ae , LLC), dating from 2015-2022.\n\n# Binomial Factoring Exercises (Problems 4-11)\n\n- Problem 4: $4x^2 - 36$\n - Identify the GCF: Both terms are divisible by $4$. Factor out the GCF to get $4(x^2 - 9)$.\n - Apply Difference of Squares: The term $(x^2 - 9)$ is a difference of squares where $a = x$ and $b = 3$.\n - Final Factored Form: $4(x - 3)(x + 3)$\n\n- Problem 5: $c^4 + 64$ (referred to in transcript as \"364 1 c +\")\n - Recognition: This is a sum of terms involving a variable raised to the fourth power. Note: Sum of squares is not factorable over real numbers, but specific cases or higher powers may involve complex identities.\n\n- Problem 6: $k^3 - 27$\n - Apply Difference of Cubes: Here, $a = k$ and $b = 3$.\n - Final Factored Form: $(k - 3)(k^2 + 3k + 9)$\n\n- Problem 7: $54x^3 + 250y^3$\n - Identify the GCF: Both coefficients are even. Factor out a $2$ to get $2(27x^3 + 125y^3)$.\n - Apply Sum of Cubes: Inside the parentheses, $a = 3x$ and $b = 5y$.\n - Final Factored Form: $2(3x + 5y)(9x^2 - 15xy + 25y^2)$\n\n- Problem 8: $48m^4n^2 - 3n^4$\n - Identify the GCF: Factor out $3n^2$ to get $3n^2(16m^4 - n^2)$.\n - Apply Difference of Squares: The term $(16m^4 - n^2)$ has $a = 4m^2$ and $b = n$.\n - Final Factored Form: $3n^2(4m^2 - n)(4m^2 + n)$\n\n- Problem 9: $a^7b^2 - ab^2$\n - Identify the GCF: Factor out $ab^2$ to get $ab^2(a^6 - 1)$.\n - Factors of $(a^6 - 1)$: This can be treated as a difference of squares $(a^3)^2 - 1^2$, resulting in $(a^3 - 1)(a^3 + 1)$.\n - Apply Cubes Formulas: Factor both resultants using Difference and Sum of Cubes.\n - Final Factored Form: $ab^2(a - 1)(a^2 + a + 1)(a + 1)(a^2 - a + 1)$\n\n- Problem 10: $5x^3y^2 - 343y^2$\n - Identify the GCF: Factor out $y^2$ to get $(5x^3 - 343)y^2$.\n\n- Problem 11: $y^9 - 144y^7$\n - Identify the GCF: Factor out $y^7$ to get $y^7(y^2 - 144)$.\n - Apply Difference of Squares: $a = y$, $b = 12$.\n - Final Factored Form: $y^7(y - 12)(y + 12)$\n\n# Trinomial and Quadratic Form Factoring (Problems 12-21)\n\n- Problem 12: $w^4 - 13w^2 + 36$\n - Substitution: Let $u = w^2$. The equation becomes $u^2 - 13u + 36$.\n - Factor the Trinomial: $(u - 9)(u - 4)$.\n - Back-substitute and Factor Completely: $(w^2 - 9)(w^2 - 4) = (w - 3)(w + 3)(w - 2)(w + 2)$.\n\n- Problem 13: $p^3 + 5p^2 - 84p$\n - Identify the GCF: Factor out $p$ to get $p(p^2 + 5p - 84)$.\n - Factor the Trinomial: Find two numbers that multiply to $-84$ and add to $5$ ($12$ and $-7$).\n - Final Factored Form: $p(p + 12)(p - 7)$\n\n- Problem 14: $x^4 - 12x^2 + 36$\n - Perfect Square Trinomial: This takes the form $(x^2 - 6)^2$.\n\n- Problem 15: $w^4 - 14w^2 - 32$\n - Factor as a Quadratic: $(w^2 - 16)(w^2 + 2)$.\n - Factor Completely: $(w - 4)(w + 4)(w^2 + 2)$.\n\n- Problem 16: $k^3 + 7k^2 - 44k$\n - Identify GCF: $k(k^2 + 7k - 44)$.\n - Factor Trinomial: $k(k + 11)(k - 4)$.\n\n- Problem 17: $a^3 + 28a^2 + 96a$\n - Identify GCF: $a(a^2 + 28a + 96)$.\n - Factor Trinomial: Numbers adding to $28$ and multiplying to $96$ are $24$ and $4$.\n - Final Factored Form: $a(a + 24)(a + 4)$\n\n- Problem 18: $-x^3 + 4x^2 + 21x$\n - Identify GCF: Factor out $-x$ to get $-x(x^2 - 4x - 21)$.\n - Factor Trinomial: $-x(x - 7)(x + 3)$.\n\n- Problem 19: $m^6 - 7m^4 - 18m^2$\n - Identify GCF: $m^2(m^4 - 7m^2 - 18)$.\n - Quadratic Form: $m^2(m^2 - 9)(m^2 + 2)$.\n - Factor Completely: $m^2(m - 3)(m + 3)(m^2 + 2)$.\n\n- Problem 20: $9y^6 + 6y^4 + y^2$\n - Identify GCF: $y^2(9y^4 + 6y^2 + 1)$.\n - Perfect Square Trinomial: $y^2(3y^2 + 1)^2$.\n\n- Problem 21: $8c^4 + 10c^2 - 3$\n - Factoring by decomposition/AC method: Find terms multiplying to $-24$ and adding to $10$ ($12$, $-2$).\n - Intermediate step: $8c^4 + 12c^2 - 2c^2 - 3 = 4c^2(2c^2 + 3) - 1(2c^2 + 3)$. \n - Final Factored Form: $(4c^2 - 1)(2c^2 + 3)$.\n - Factored Completely: $(2c - 1)(2c + 1)(2c^2 + 3)$.\n\n# Factoring by Grouping (Problems 22-25)\n\n- Problem 22: $x^3 - 7x^2 + x - 7$\n - Grouping groups: $(x^3 - 7x^2) + (x - 7)$.\n - Factoring: $x^2(x - 7) + 1(x - 7)$.\n - Final Factored Form: $(x^2 + 1)(x - 7)$\n\n- Problem 23: $4r^3 - 3r^2 - 4r + 3$\n - Grouping groups: $(4r^3 - 3r^2) - (4r - 3)$.\n - Factoring: $r^2(4r - 3) - 1(4r - 3)$.\n - Intermediate step: $(r^2 - 1)(4r - 3)$.\n - Factor Completely: $(r - 1)(r + 1)(4r - 3)$\n\n- Problem 24: $3p^3 + 5p^2 - 12p - 20$\n - Grouping groups: $p^2(3p + 5) - 4(3p + 5)$.\n - Intermediate step: $(p^2 - 4)(3p + 5)$.\n - Factor Completely: $(p - 2)(p + 2)(3p + 5)$\n\n- Problem 25: $15n^3 - 6n^2 - 25n + 10$\n - Grouping groups: $3n^2(5n - 2) - 5(5n - 2)$.\n - Final Factored Form: $(3n^2 - 5)(5n - 2)$", "title": "Unit 5: Polynomial Functions - Factoring Polynomials Study Guide"}