Algebra Factorization Techniques
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Questions and Answers

What is the factorization of $a^4 - b^4$?

  • (a^2 - b^2)(a^2 + b^2) (correct)
  • (a - b)(a + b)(a^2 + b^2)
  • (a^2 + b^2)(a^2 - b^2)
  • (a^2 - 2ab + b^2)(a^2 + 2ab + b^2)

    • Which of the following represents the correct factorization of $p^4 - 81$?

  • (p^2 - 3)(p^2 + 27)
  • (p^2 - 9)(p^2 + 9) (correct)
  • (p^2 - 9)(p^2 + 3)
  • (p - 3)(p + 3)(p^2 + 9)

    • What is the correct factorization for the expression $x^4 - (y + z)^4$?

  • $(x^2 - y^2 - 2yz - z^2)(x^2 + y^2 + 2yz + z^2)$
  • $(x - (y + z))(x + (y + z))(x^2 + (y + z)^2)$
  • $(x^2 - (y + z))(x^2 + (y + z))$
  • (x^2 - (y + z)^2)(x^2 + (y + z)^2) (correct)

    • Which factorization accurately represents $x^4 - (x - z)^4$?

    <p>$(x^2 - (x - z)^2)(x^2 + (x - z)^2)$</p> Signup and view all the answers

    What is the correct factorization of $a^4 - 2a^2b^2 + b^4$?

    <p>$(a^2 - b^2)^2$</p> Signup and view all the answers

    What is the factorization of the expression $x^2 + xy + 8x + 8y$?

    <p>$(x + y)(x + 8)$</p> Signup and view all the answers

    Factor the expression $15xy - 6x + 5y - 2$.

    <p>$(5y - 2)(3x + 1)$</p> Signup and view all the answers

    Which of the following represents the factorization of $ax + bx - ay - by$?

    <p>$x(a + b) - y(a + b)$</p> Signup and view all the answers

    What is the correct factorization of the expression $15pq + 9q + 25p + 15$?

    <p>$(3q + 5)(5p + 3)$</p> Signup and view all the answers

    When factoring $z - 7 + 7xy - xyz$, which of the following is correct?

    <p>$(1 - xy)(z - 7)$</p> Signup and view all the answers

    What is the factorization of the expression $a^2 + 8a + 16$?

    <p>$(a + 4)^2$</p> Signup and view all the answers

    Which expression is correctly factored: $p^2 - 10p + 25$?

    <p>$(p - 5)^2$</p> Signup and view all the answers

    What is the factorization of the expression $121b^2 - 88bc + 16c^2$?

    <p>$(11b - 4c)^2$</p> Signup and view all the answers

    What is the factored form of the expression $3y^8 - 4y^6 + 5y^4$?

    <p>$y^4(3y^4 - 4y^2 + 5)$</p> Signup and view all the answers

    What is the result of dividing the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ by $4xy^2z^2$?

    <p>$2(x + y + z)$</p> Signup and view all the answers

    What expression results from the division of $(x^3 + 2x^2 + 3x)$ by $2x$?

    <p>$ rac{1}{2}x^2 + x + rac{3}{2}$</p> Signup and view all the answers

    What common factor can be removed from the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ before division?

    <p>$2x^2y^2z^2$</p> Signup and view all the answers

    What is the effect of dividing the expression $3y^4 - 4y^2 + 5$ by $y^4$?

    <p>$3 - rac{4}{y^2} + rac{5}{y^4}$</p> Signup and view all the answers

    What is the correct factorization of the expression $12xy(9x^2 - 16y^2)$ using the identity $a^2 - b^2 = (a - b)(a + b)$?

    <p>$12xy(3x - 4y)(3x + 4y)$</p> Signup and view all the answers

    What is the result after canceling common factors in the expression $\frac{12xy(9x^2 - 16y^2)}{4xy(3x + 4y)}$?

    <p>$3(3x - 4y)$</p> Signup and view all the answers

    Which identity is used to factor the expression $39y^3(50y^2 - 98)$?

    <p>$a^2 - b^2 = (a - b)(a + b)$</p> Signup and view all the answers

    What is the correct simplification of $\frac{39y^3(50y^2 - 98)}{26y^2(5y + 7)}$?

    <p>$3y(5y - 7)$</p> Signup and view all the answers

    What is the error in the expression $4(x - 5) = 4x - 5$?

    <p>It incorrectly distributes across the subtraction.</p> Signup and view all the answers

    What correction can be made to the statement $x(3x + 2) = 3x^2 + 2$?

    <p>Change $3x^2 + 2$ to $3x^2 + 2x$.</p> Signup and view all the answers

    Which expression represents the correct factorization of $50y^2 - 98$?

    <p>$2(5y - 7)(5y + 7)$</p> Signup and view all the answers

    How is the expression $39y^3(50y^2 - 98)$ rewritten for factorization?

    <p>$3 imes 13 imes 2 imes y^3 imes (5y - 7)(5y + 7)$</p> Signup and view all the answers

    What is the factored form of $a^4 - b^4$?

    <p>$(a - b)(a + b)(a^2 + b^2)$</p> Signup and view all the answers

    Which expression represents the factorization of $p^4 - 81$?

    <p>$(p^2 - 9)(p^2 + 9)$</p> Signup and view all the answers

    What is the correct factorization for the expression $x^4 - (y + z)^4$?

    <p>$(x^2 - (y + z)^2)(x^2 + (y + z)^2)$</p> Signup and view all the answers

    How can you simplify the expression $x^4 - (x - z)^4$?

    <p>$(z)(2x - z)(x^2 + x^2 - 2xz + z^2)$</p> Signup and view all the answers

    What is the factored form of $p^2 + 6p + 8$?

    <p>$(p + 4)(p + 2)$</p> Signup and view all the answers

    What is the correct factorization of $q^2 - 10q + 21$?

    <p>$(q - 7)(q - 3)$</p> Signup and view all the answers

    Which of the following is the factored representation of $p^2 + 6p - 16$?

    <p>$(p - 2)(p + 8)$</p> Signup and view all the answers

    Which expression corresponds with the identity $(a - b)^2 = a^2 - 2ab + b^2$?

    <p>$a^2 - 2ab + b^2$</p> Signup and view all the answers

    Study Notes

    Factorization of Expressions

    • The expression ax^2y + bxy^2 + cxyz can be factored as xy(ax + by + cz).

    • Factorizing by Grouping:

      • Group terms with common factors.
      • Factor out the common factors from each group.
      • Factor out the common binomial factor from the resulting two terms.

    • Examples of Factorizing by Grouping:

      • x^2 + xy + 8x + 8y = (x + y)(x + 8)
      • 15xy - 6x + 5y - 2 = (5y - 2)(3x + 1)
      • ax + bx - ay - by = (a + b)(x - y)
      • 15pq + 15 + 9q + 25p = (5p + 3)(3q + 5)
      • z - 7 + 7xy - xyz = (1 - xy)(z - 7)

    Factorization of Perfect Square Trinomials

    • Perfect Square Trinomial

      • A trinomial that results from squaring a binomial.
      • In the form a^2 + 2ab + b^2 or a^2 - 2ab + b^2

    • Factoring a Perfect Square Trinomial

      • (a + b)^2 = a^2 + 2ab + b^2
      • (a - b)^2 = a^2 - 2ab + b^2

    • Examples of Factoring Perfect Square Trinomials

      • a^2 + 8a + 16 = (a + 4)^2
      • p^2 - 10p + 25 = (p - 5)^2
      • 25m^2 + 30m + 9 = (5m + 3)^2
      • 49y^2 + 84yz + 36z^2 = (7y + 6z)^2
      • 4x^2 - 8x + 4 = (2x - 2)^2
      • 121b^2 - 88bc + 16c^2 = (11b - 4c)^2
      • (l + m)^2 - 4lm = (l - m)^2
      • a^4 + 2a^2b^2 + b^4 = (a^2 + b^2)^2

    Division of Algebraic Expressions

    • Dividing expressions with factored terms

      • Factorize both the numerator and denominator.
      • Cancel out common factors.

    • Examples of Dividing Algebraic Expressions

      • 12xy(9x^2 - 16y^2) ÷ 4xy(3x + 4y) = 3(3x - 4y)
      • 39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) = 3y(5y - 7)

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    Description

    This quiz focuses on various factorization techniques in algebra, including factorizing expressions by grouping and perfect square trinomials. You'll explore different methods and examples, enhancing your understanding of polynomial identities and factoring skills.

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