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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 $(a - b)^2 = a^2 - 2ab + b^2$?
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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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