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Kuta Software Factoring Vocabulary

Factorization Practice Notes (Math 2: 2.4-2.5)

  • These notes summarize and organize the in-class practice problems on factoring from the transcript. They cover:

    • Factoring quadratics (trinomials with a, b, c)
    • Factoring with a GCF (greatest common factor)
    • Factoring by grouping
    • Solutions and final factored forms for all problems listed
    • Key strategies and common pitfalls

  • Important concepts to review:

    • Zero Product Property: If a product equals zero, at least one factor must be zero. Factoring is used to find roots.
    • For quadratics: look for two numbers that multiply to ac and add to b (for ax^2 + bx + c form). When a ≠ 1, divide and factor or find factor pairs that work with a common factor approach.
    • GCF first: Always factor out the greatest common factor before attempting further factoring.
    • Grouping method: Group terms in pairs to pull out common factors and produce a binomial factor common to both groups.
    • If a polynomial does not factor over the integers, indicate that it factors as a constant times an irreducible quadratic or leave in factored-with-GCF form.

Factor by taking out the GCF first (problems 7–12)

  • General approach: Factor out the greatest common factor (GCF) from the entire polynomial, then factor the remaining polynomial if possible.

  • 7) Expression: 3x^3 + 12x^2 - 135x

    • Step 1: Factor out GCF: 3x
    • Step 2: Inside remaining: x^2 + 4x - 45
    • Step 3: Factor the quadratic: numbers that multiply to -45 and add to 4 are 9 and -5 (but note signs; proper decomposition yields factors (x+9)(x-5))
    • Full factorization: 3x(x+9)(x-5)
    • Verification: The product expands to the original expression.

  • 8) Expression: 2k^3 - 10k^2 - 12k

    • Step 1: GCF is 2k: 2k(k^2 - 5k - 6)
    • Step 2: Factor the quadratic: two numbers multiply to -6 and add to -5 are -6 and +1, giving
      k^2 - 5k - 6 = (k-6)(k+1)
    • Full factorization: 2k(k-6)(k+1)

  • 9) Expression: 4p^2 - 80p - 360

    • Step 1: GCF is 4: 4(p^2 - 20p - 90)
    • Step 2: Check if the inner quadratic factors over integers: Discriminant = (-20)^2 - 4(1)(-90) = 400 + 360 = 760, not a perfect square, so it does not factor over the integers.
    • Conclusion: Fully factored over integers is 4(p^2 - 20p - 90); if allowed, one could write roots using the quadratic formula, but it’s not a clean integer factorization.

  • 10) Expression: 4x^4 + 16x^3 - 20x^2

    • Step 1: GCF is 4x^2: 4x^2(x^2 + 4x - 5)
    • Step 2: Factor the quadratic: x^2 + 4x - 5 = (x+5)(x-1)
    • Full factorization: 4x^2(x+5)(x-1)

  • 11) Expression: 5x^2 - 35x - 315

    • Step 1: GCF is 5: 5(x^2 - 7x - 63)
    • Step 2: Check factoring of the inner quadratic: Discriminant (-7)^2 - 4(1)(-63) = 49 + 252 = 301 (not a perfect square)
    • Conclusion: Cannot factor over integers beyond the GCF; final form is 5(x^2 - 7x - 63).

  • 12) Expression: 5a^2 + 5a - 30

    • Step 1: GCF is 5: 5(a^2 + a - 6)
    • Step 2: Factor the inner quadratic: find two numbers that multiply to -6 and add to +1: 3 and -2, so a^2 + a - 6 = (a+3)(a-2)
    • Full factorization: 5(a+3)(a-2)

Factor by grouping (problems 13–24)

  • Key idea: Group the terms into two pairs so that each pair has a common factor, then factor out the common binomial factor from the two groups.

  • Note: If necessary, you can factor a negative from the right-hand group to reveal a common binomial factor.

  • 13) Expression: 18n^2 + 48n + 15n + 40

    • Grouping: $(18n^2 + 48n) + (15n + 40)$
    • Factor each group: 6n(3n+8) + 5(3n+8)
    • Common binomial factor: (3n+8)(6n+5)

  • 14) Expression: 24r^2 + 64r + 15r + 40

    • Grouping: $(24r^2 + 64r) + (15r + 40)$
    • Factor: 8r(3r+8) + 5(3r+8)
    • Final: (3r+8)(8r+5)

  • 15) Expression: 14n^2 + 35n + 4n + 10

    • Grouping: $(14n^2 + 35n) + (4n + 10)$
    • Factor: 7n(2n+5) + 2(2n+5)
    • Final: (2n+5)(7n+2)

  • 16) Expression: 6n^2 + 7n + 48n + 56

    • Grouping: $(6n^2 + 7n) + (48n + 56)$
    • Factor: n(6n+7) + 8(6n+7)
    • Final: (6n+7)(n+8)

  • 17) Expression: 5n^2 + 35n + 6n + 42

    • Grouping: $(5n^2 + 35n) + (6n + 42)$
    • Factor: 5n(n+7) + 6(n+7)
    • Final: (n+7)(5n+6)

  • 18) Expression: 5x^2 - 25x + x - 5

    • Grouping: $(5x^2 - 25x) + (x - 5)$
    • Factor: 5x(x-5) + 1(x-5)
    • Final: (5x+1)(x-5)

  • 19) Expression: 30m^2 - 18m - 25m + 15

    • Grouping: $(30m^2 - 18m) + (-25m + 15)$
    • Factor: 6m(5m-3) -5(5m-3)
    • Final: (5m-3)(6m-5)

  • 20) Expression: 18x^2 + 30x - 15x - 25

    • Grouping: $(18x^2 + 30x) + (-15x - 25)$
    • Factor: 6x(3x+5) -5(3x+5)
    • Final: (3x+5)(6x-5)

  • 21) Expression: 28k^2 + 7k - 12k - 3

    • Grouping: $(28k^2 + 7k) + (-12k - 3)$
    • Factor: 7k(4k+1) -3(4k+1)
    • Final: (4k+1)(7k-3)

  • 22) Expression: 4x^2 - 7x - 8x + 14

    • Grouping: $(4x^2 - 7x) + (-8x + 14)$
    • Factor: x(4x-7) -2(4x-7)
    • Final: (4x-7)(x-2)

  • 23) Expression: 28n^2 - 32n - 21n + 24

    • Grouping: $(28n^2 - 32n) + (-21n + 24)$
    • Factor: 4n(7n-8) -3(7n-8)
    • Final: (7n-8)(4n-3)

  • 24) Expression: 5m^2 - 4m - 40m + 32

    • Grouping: $(5m^2 - 4m) + (-40m + 32)$
    • Factor: m(5m-4) -8(5m-4)
    • Final: (5m-4)(m-8)

Quick reference: patterns and tips

  • Quadratic trinomials ax^2 + bx + c (a ≠ 0): look for two numbers that multiply to a·c and add to b. If a = 1, this reduces to simply finding two numbers that multiply to c and add to b (for monic quadratics).
  • When factoring by grouping:
    • Group terms so that each group has a common factor.
    • Extract the common factors from each group.
    • If a common binomial factor emerges, factor it out to complete the factorization.
    • If grouping does not yield an immediate common binomial factor, check for alternate groupings or apply alternative methods (GCF, special products).
  • Always verify by expanding the factors to ensure they reproduce the original polynomial.

Connections to broader topics

  • Factoring is a fundamental step in solving polynomial equations, simplifying expressions, and understanding polynomial behavior (zeros, intercepts).
  • Factoring by grouping is especially useful for higher-degree polynomials that can be decomposed into two binomial factors after grouping.
  • The zero-product property ties factoring techniques to solving equations: if a product of factors equals zero, each factor equated to zero gives roots.

Real-world relevance and study tips

  • Factoring shows up in physics (e.g., simplifying expressions in kinematics and energy equations), coding and computer algebra systems, and engineering applications that require solving polynomial constraints.
  • Practice tip: for each factoring problem, first check for a GCF; if not present, look for two-binomial groupings or known factor patterns (difference of squares, perfect square trinomials, sum/difference of cubes where applicable).
  • For problems like 9) where the inner quadratic does not factor over integers, note the limitation and keep the factorization at the GCF stage unless a non-integer factorization is acceptable for the context.

Summary of all problem results (concise list)

  • 1) n^2 - 10n + 24 = (n-6)(n-4)

  • 2) n^2 + 3n - 54 = (n+9)(n-6)

  • 3) m^2 - 17m + 72 = (m-9)(m-8)

  • 4) x^2 - 11x + 24 = (x-8)(x-3)

  • 5) p^2 + 3p - 54 = (p+9)(p-6)

  • 6) x^2 + 10x + 16 = (x+2)(x+8)

  • 7) 3x^3 + 12x^2 - 135x = 3x(x+9)(x-5)

  • 8) 2k^3 - 10k^2 - 12k = 2k(k-6)(k+1)

  • 9) 4p^2 - 80p - 360 = 4(p^2 - 20p - 90) (no integer factorization)

  • 10) 4x^4 + 16x^3 - 20x^2 = 4x^2(x+5)(x-1)

  • 11) 5x^2 - 35x - 315 = 5(x^2 - 7x - 63) (irreducible over integers further)

  • 12) 5a^2 + 5a - 30 = 5(a+3)(a-2)

  • 13) 18n^2 + 48n + 15n + 40 = (3n+8)(6n+5)

  • 14) 24r^2 + 64r + 15r + 40 = (3r+8)(8r+5)

  • 15) 14n^2 + 35n + 4n + 10 = (2n+5)(7n+2)

  • 16) 6n^2 + 7n + 48n + 56 = (6n+7)(n+8)

  • 17) 5n^2 + 35n + 6n + 42 = (n+7)(5n+6)

  • 18) 5x^2 - 25x + x - 5 = (5x+1)(x-5)

  • 19) 30m^2 - 18m - 25m + 15 = (5m-3)(6m-5)

  • 20) 18x^2 + 30x - 15x - 25 = (3x+5)(6x-5)

  • 21) 28k^2 + 7k - 12k - 3 = (4k+1)(7k-3)

  • 22) 4x^2 - 7x - 8x + 14 = (4x-7)(x-2)

  • 23) 28n^2 - 32n - 21n + 24 = (7n-8)(4n-3)

  • 24) 5m^2 - 4m - 40m + 32 = (5m-4)(m-8)

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