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Questions and Answers
What is the factorization of $a^4 - b^4$?
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 $p^4 - 81$?
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 $x^4 - (y + z)^4$?
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 $x^4 - (x - z)^4$?
Which factorization accurately represents $x^4 - (x - z)^4$?
What is the correct factorization of $a^4 - 2a^2b^2 + b^4$?
What is the correct factorization of $a^4 - 2a^2b^2 + b^4$?
What is the factorization of the expression $x^2 + xy + 8x + 8y$?
What is the factorization of the expression $x^2 + xy + 8x + 8y$?
Factor the expression $15xy - 6x + 5y - 2$.
Factor the expression $15xy - 6x + 5y - 2$.
Which of the following represents the factorization of $ax + bx - ay - by$?
Which of the following represents the factorization of $ax + bx - ay - by$?
What is the correct factorization of the expression $15pq + 9q + 25p + 15$?
What is the correct factorization of the expression $15pq + 9q + 25p + 15$?
When factoring $z - 7 + 7xy - xyz$, which of the following is correct?
When factoring $z - 7 + 7xy - xyz$, which of the following is correct?
What is the factorization of the expression $a^2 + 8a + 16$?
What is the factorization of the expression $a^2 + 8a + 16$?
Which expression is correctly factored: $p^2 - 10p + 25$?
Which expression is correctly factored: $p^2 - 10p + 25$?
What is the factorization of the expression $121b^2 - 88bc + 16c^2$?
What is the factorization of the expression $121b^2 - 88bc + 16c^2$?
What is the factored form of the expression $3y^8 - 4y^6 + 5y^4$?
What is the factored form of the expression $3y^8 - 4y^6 + 5y^4$?
What is the result of dividing the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ by $4xy^2z^2$?
What is the result of dividing the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ by $4xy^2z^2$?
What expression results from the division of $(x^3 + 2x^2 + 3x)$ by $2x$?
What expression results from the division of $(x^3 + 2x^2 + 3x)$ by $2x$?
What common factor can be removed from the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ before division?
What common factor can be removed from the expression $8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3)$ before division?
What is the effect of dividing the expression $3y^4 - 4y^2 + 5$ by $y^4$?
What is the effect of dividing the expression $3y^4 - 4y^2 + 5$ by $y^4$?
What is the correct factorization of the expression $12xy(9x^2 - 16y^2)$ using the identity $a^2 - b^2 = (a - b)(a + b)$?
What is the correct factorization of the expression $12xy(9x^2 - 16y^2)$ using the identity $a^2 - b^2 = (a - b)(a + b)$?
What is the result after canceling common factors in the expression $\frac{12xy(9x^2 - 16y^2)}{4xy(3x + 4y)}$?
What is the result after canceling common factors in the expression $\frac{12xy(9x^2 - 16y^2)}{4xy(3x + 4y)}$?
Which identity is used to factor the expression $39y^3(50y^2 - 98)$?
Which identity is used to factor the expression $39y^3(50y^2 - 98)$?
What is the correct simplification of $\frac{39y^3(50y^2 - 98)}{26y^2(5y + 7)}$?
What is the correct simplification of $\frac{39y^3(50y^2 - 98)}{26y^2(5y + 7)}$?
What is the error in the expression $4(x - 5) = 4x - 5$?
What is the error in the expression $4(x - 5) = 4x - 5$?
What correction can be made to the statement $x(3x + 2) = 3x^2 + 2$?
What correction can be made to the statement $x(3x + 2) = 3x^2 + 2$?
Which expression represents the correct factorization of $50y^2 - 98$?
Which expression represents the correct factorization of $50y^2 - 98$?
How is the expression $39y^3(50y^2 - 98)$ rewritten for factorization?
How is the expression $39y^3(50y^2 - 98)$ rewritten for factorization?
What is the factored form of $a^4 - b^4$?
What is the factored form of $a^4 - b^4$?
Which expression represents the factorization of $p^4 - 81$?
Which expression represents the factorization of $p^4 - 81$?
What is the correct factorization for the expression $x^4 - (y + z)^4$?
What is the correct factorization for the expression $x^4 - (y + z)^4$?
How can you simplify the expression $x^4 - (x - z)^4$?
How can you simplify the expression $x^4 - (x - z)^4$?
What is the factored form of $p^2 + 6p + 8$?
What is the factored form of $p^2 + 6p + 8$?
What is the correct factorization of $q^2 - 10q + 21$?
What is the correct factorization of $q^2 - 10q + 21$?
Which of the following is the factored representation of $p^2 + 6p - 16$?
Which of the following is the factored representation of $p^2 + 6p - 16$?
Which expression corresponds with the identity $(a - b)^2 = a^2 - 2ab + b^2$?
Which expression corresponds with the identity $(a - b)^2 = a^2 - 2ab + b^2$?
Study Notes
Factorization of Expressions
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The expression ax^2y + bxy^2 + cxyz can be factored as xy(ax + by + cz).
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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.
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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
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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
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Factoring a Perfect Square Trinomial
- (a + b)^2 = a^2 + 2ab + b^2
- (a - b)^2 = a^2 - 2ab + b^2
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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
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Dividing expressions with factored terms
- Factorize both the numerator and denominator.
- Cancel out common factors.
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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.
