Factoring Out Greatest Common Factor (GCF) - College Algebra

Which shows $8x^3+4x$ factored completely by its GCF?
Answer: $4x(2x^2+1)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $8x^3$ + 4x, the GCF is 4x, which, when factored out, results in the expression 4x( $2x^2$ + 1). The correct choice, 4x( $2x^2+1$ ), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor 'x( $8x^2+4$ )', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which of the following shows $24x^2-36x$ factored completely by its GCF?
Answer: $12x(2x-3)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $24x^2$ - 36x, the GCF is 12x, which, when factored out, results in the expression 12x(2x - 3). The correct choice, 12x(2x - 3), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '6x(4x - 6)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Factor out the GCF: $4xy+8x^2y^2$ .
Answer: $4xy(1+2xy)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, 4x y + $8x^2$ $y^2$ , the GCF is 4x y, which, when factored out, results in the expression 4x y(1 + 2x y). The correct choice, 4x y(1 + 2x y), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '4x(y + 2x $y^2$ )', which misses the common y in the second term properly. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which shows $18x^3y+12x^2y^2$ factored completely by its GCF?
Answer: $6x^2y(3x+2y)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $18x^3$ y + $12x^2$ $y^2$ , the GCF is $6x^2$ y, which, when factored out, results in the expression $6x^2$ y(3x + 2y). The correct choice, $6x^2$ y(3x+2y), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor ' $3x^2$ y(6x+4y)', which misses the largest common factor. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which expression represents the complete factorization of $24x^3y^2 - 18x^2y^3 + 30xy^4$ ?
Answer: $6xy^2(4x^2 - 3xy + 5y^2)$ First, find the GCF of the coefficients: GCF(24, 18, 30) = 6. For variables, the lowest power of x is x¹ and the lowest power of y is y². So GCF = 6xy². Factoring out: 24x³y² ÷ 6xy² = 4x², 18x²y³ ÷ 6xy² = 3xy, 30xy⁴ ÷ 6xy² = 5y². Choice B incorrectly leaves xy instead of xy²; Choice C uses 3 instead of 6 as the coefficient; Choice D uses 12 instead of 6, creating fractions.
How can $-12x^2-18x$ be simplified by factoring out the GCF?
Answer: $-6x(2x+3)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, - $12x^2$ - 18x, the GCF is -6x (or equivalently 6x with negatives inside), which, when factored out, results in the expression -6x(2x + 3). The correct choice, -6x(2x+3), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is ignoring the negative sign, seen in distractors that do not match the original expression. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which expression shows $15xy+20y$ factored completely by its GCF?
Answer: $5y(3x+4)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, 15x y + 20 y, the GCF is 5 y, which, when factored out, results in the expression 5 y(3x + 4). The correct choice, 5 y(3x + 4), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor 'y(15x + 20)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which of the following shows $6x^2y+9xy$ factored by its GCF?
Answer: $3xy(2x+3)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $6x^2$ y + 9x y, the GCF is 3x y, which, when factored out, results in the expression 3x y(2x + 3). The correct choice, 3x y(2x + 3), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor 'x y(6x + 9)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Factor out the GCF: $45ab-30a$ .
Answer: $15a(3b-2)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, 45a b - 30a, the GCF is 15a, which, when factored out, results in the expression 15a(3b - 2). The correct choice, 15a(3b - 2), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '5a(9b - 6)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Identify the GCF and rewrite: $16m^2n-24mn^2$ .
Answer: $8mn(2m-3n)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $16m^2$ n - 24m $n^2$ , the GCF is 8m n, which, when factored out, results in the expression 8m n(2m - 3n). The correct choice, 8m n(2m-3n), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor ' $8m^2$ n(2-3n)', which misses the correct variable powers. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
When factoring $21x^4y^3 + 14x^3y^2 - 35x^2y^4$ , a student writes $7x^2y^2(3x^2y + 2x - 5y^2)$ . What error, if any, did the student make?
Answer: No error was made; the factorization is correct Let's verify the student's work. The GCF of coefficients 21, 14, 35 is 7. For variables: lowest power of x is x² and lowest power of y is y², so GCF = 7x²y². Now check the factoring: 21x⁴y³ ÷ 7x²y² = 3x²y, 14x³y² ÷ 7x²y² = 2x, -35x²y⁴ ÷ 7x²y² = -5y². Verification: 7x²y²(3x²y + 2x - 5y²) = 21x⁴y³ + 14x³y² - 35x²y⁴ ✓. The student's factorization is completely correct.
A rectangular garden has area represented by $15x^3 + 10x^2 - 25x$ square feet. If the width is the GCF of the terms in this expression, what polynomial represents the length?
Answer: $3x^2 + 2x - 5$ When you see a polynomial representing area and need to find dimensions, you're working with polynomial factoring. Since area = length × width, you need to factor the area expression to find the individual dimensions. First, find the GCF (Greatest Common Factor) of $15x^3 + 10x^2 - 25x$ . Look at both the coefficients and variables: - Coefficients: 15, 10, and 25 all share a common factor of 5 - Variables: Each term contains at least one factor of $x$ So the GCF is $5x$ , which represents the width. To find the length, divide the area by the width: $\frac{15x^3 + 10x^2 - 25x}{5x} = \frac{15x^3}{5x} + \frac{10x^2}{5x} - \frac{25x}{5x} = 3x^2 + 2x - 5$ This confirms answer C is correct. Let's examine why the other choices are wrong: A) $3x^3 + 2x^2 - 5x$ incorrectly assumes the GCF is just 5, not $5x$ . This would give a width of 5, but that ignores the variable component. B) $15x^2 + 10x - 25$ assumes the GCF is only $x$ , missing the coefficient factor of 5. D) $5x^2 + 2x - 5$ results from incorrectly dividing only some terms by the full GCF—a common algebraic error. Study tip: When factoring polynomials, always check both numerical coefficients AND variable parts for the GCF. Then verify your answer by multiplying the factors back together to ensure you get the original expression.
Which expression shows $27x^3y-9x^2y$ factored by its GCF?
Answer: $9x^2y(3x-1)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $27x^3$ y - $9x^2$ y, the GCF is $9x^2$ y, which, when factored out, results in the expression $9x^2$ y(3x - 1). The correct choice, $9x^2$ y(3x - 1), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '9x y( $3x^2$ - x)', which misses the full variable power in the GCF. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Factor out the GCF: $14x^3-21x^2$ .
Answer: $7x^2(2x-3)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $14x^3$ - $21x^2$ , the GCF is $7x^2$ , which, when factored out, results in the expression $7x^2$ (2x - 3). The correct choice, $7x^2$ (2x - 3), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '7x( $2x^2$ - 3x)', which misses the full variable power in the GCF. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Identify the GCF and factor: $18a^2b+24ab^2$ .
Answer: $6ab(3a+4b)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $18a^2$ b + 24a $b^2$ , the GCF is 6a b, which, when factored out, results in the expression 6a b(3a + 4b). The correct choice, 6a b(3a + 4b), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor 'a b(18a + 24b)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Choose the correct GCF for $24a^2b$ and $36ab^2$ , then factor it out.
Answer: $12ab(2a+3b)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given terms, $24a^2$ b and 36a $b^2$ , the GCF is 12ab, which, when factored out, results in the expression 12ab(2a + 3b). The correct choice, 12ab(2a+3b), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '6ab(4a+6b)', which misses the largest common factor. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Which of the following shows $2x^2y+6xy^2$ factored by its GCF?
Answer: $2xy(x+3y)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, ( $2x^2$ y + 6x $y^2$ ), the GCF is (2xy), which, when factored out, results in the expression (2xy(x + 3y)). The correct choice, (2xy(x + 3y)), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in distractors like (xy(2x + 6y)) or (2y( $x^2$ + 3xy)), which miss the largest common factor. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
How can $15xy+20y$ be simplified by factoring out the GCF?
Answer: $5y(3x+4)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, 15xy + 20y, the GCF is 5y, which, when factored out, results in the expression 5y(3x + 4). The correct choice, 5y(3x+4), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor 'y(15x+20)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Identify the GCF and factor: $6x^4+9x^3$ .
Answer: $3x^3(2x+3)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $6x^4$ + $9x^3$ , the GCF is $3x^3$ , which, when factored out, results in the expression $3x^3$ (2x + 3). The correct choice, $3x^3$ (2x + 3), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '3x( $2x^3$ + $3x^2$ )', which misses the full variable power in the GCF. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.
Identify the GCF and factor: $54x^2y-18xy$ .
Answer: $18xy(3x-1)$ This question tests college-level algebra skills in factoring out the Greatest Common Factor (GCF). The concept involves identifying the largest factor shared among terms and using it to simplify expressions. In the given expression, $54x^2$ y - 18x y, the GCF is 18x y, which, when factored out, results in the expression 18x y(3x - 1). The correct choice, 18x y(3x - 1), accurately represents the factorization, demonstrating understanding of the GCF concept. A common mistake is partial factorization, seen in the distractor '9x y(6x - 2)', which misses the largest common coefficient. To support students, emphasize practice with identifying GCF through various expressions, and reinforce the importance of checking work for complete factorization. Encourage exercises in different contexts to strengthen recognition of common factors.

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