Practice
Factoring Using the Distributive Property Practice
Fifty original questions covering GCF extraction, negative factors, grouping, opposite factors, verification, and error analysis.
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Question 1
Explanation
The coefficient GCF is \(6\). Dividing both terms by it gives \(3x+5\), so the complete factorization is \(6(3x+5)\).
Question 2
Explanation
The coefficient GCF is \(14\). Dividing both terms by it gives \(2x+3\), so the complete factorization is \(14(2x+3)\).
Question 3
Explanation
The coefficient GCF is \(12\). Dividing both terms by it gives \(2x-3\), so the complete factorization is \(12(2x-3)\).
Question 4
Explanation
The coefficient GCF is \(5\). Dividing both terms by it gives \(7x+10\), so the complete factorization is \(5(7x+10)\).
Question 5
Explanation
The coefficient GCF is \(22\). Dividing both terms by it gives \(2x-3\), so the complete factorization is \(22(2x-3)\).
Question 6
Explanation
Use the smallest exponent of each shared variable, then divide every term. Expanding \(x^3(x^3+1)\) reproduces \(x^6+x^3\).
Question 7
Explanation
Use the smallest exponent of each shared variable, then divide every term. Expanding \(y^5(y^3-4)\) reproduces \(y^8-4y^5\).
Question 8
Explanation
Use the smallest exponent of each shared variable, then divide every term. Expanding \(a^3b^4(a^4+b^2)\) reproduces \(a^7b^4+a^3b^6\).
Question 9
Explanation
Use the smallest exponent of each shared variable, then divide every term. Expanding \(m^2n^2(m^3-n^5)\) reproduces \(m^5n^2-m^2n^7\).
Question 10
Explanation
Use the smallest exponent of each shared variable, then divide every term. Expanding \(p^4q^3(p^5+q^5)\) reproduces \(p^9q^3+p^4q^8\).
Question 11
Explanation
The numeric GCF and minimum shared variable powers form the outside factor. Term-by-term division gives \(12x^3y^3(2x^2-3y^2)\), and distribution verifies the original expression.
Question 12
Explanation
The numeric GCF and minimum shared variable powers form the outside factor. Term-by-term division gives \(15a^4b^2(2a^2+3b^3)\), and distribution verifies the original expression.
Question 13
Explanation
The numeric GCF and minimum shared variable powers form the outside factor. Term-by-term division gives \(14m^3n^4(3m^4-2n^2)\), and distribution verifies the original expression.
Question 14
Explanation
The numeric GCF and minimum shared variable powers form the outside factor. Term-by-term division gives \(18p^3q^2(3p^2q^6+4)\), and distribution verifies the original expression.
Question 15
Explanation
The numeric GCF and minimum shared variable powers form the outside factor. Term-by-term division gives \(21r^4s^3(3r^5-4s^4)\), and distribution verifies the original expression.
Question 16
Explanation
Extracting a negative GCF reverses every inside sign. The result \(-6x^2(3x-5)\) expands exactly to \(-18x^3+30x^2\).
Question 17
Explanation
Extracting a negative GCF reverses every inside sign. The result \(-12y^3(2y^2+3)\) expands exactly to \(-24y^5-36y^3\).
Question 18
Explanation
Extracting a negative GCF reverses every inside sign. The result \(-7a^2(5a^2-7)\) expands exactly to \(-35a^4+49a^2\).
Question 19
Explanation
Extracting a negative GCF reverses every inside sign. The result \(-20m^3n(2m^3-3n)\) expands exactly to \(-40m^6n+60m^3n^2\).
Question 20
Explanation
Extracting a negative GCF reverses every inside sign. The result \(-18p^2q^2(3p^3+4q^2)\) expands exactly to \(-54p^5q^2-72p^2q^4\).
Question 21
Explanation
Group the first two and last two terms. Their common binomial is \(4x+5\), leaving \(2x+3\). Expansion verifies \((2x+3)(4x+5)=8x^2+10x+12x+15\).
Question 22
Explanation
Group the first two and last two terms. Their common binomial is \(5x+4\), leaving \(3x-2\). Expansion verifies \((3x-2)(5x+4)=15x^2+12x-10x-8\).
Question 23
Explanation
Group the first two and last two terms. Their common binomial is \(3x-6\), leaving \(4x+1\). Expansion verifies \((4x+1)(3x-6)=12x^2-24x+3x-6\).
Question 24
Explanation
Group the first two and last two terms. Their common binomial is \(2x+7\), leaving \(5x-3\). Expansion verifies \((5x-3)(2x+7)=10x^2+35x-6x-21\).
Question 25
Explanation
Group the first two and last two terms. Their common binomial is \(-3x+4\), leaving \(2x+5\). Expansion verifies \((2x+5)(-3x+4)=-6x^2+8x-15x+20\).
Question 26
Explanation
Group the first two and last two terms. Their common binomial is \(2x-5\), leaving \(6x+1\). Expansion verifies \((6x+1)(2x-5)=12x^2-30x+2x-5\).
Question 27
Explanation
Group the first two and last two terms. Their common binomial is \(4x+2\), leaving \(3x+7\). Expansion verifies \((3x+7)(4x+2)=12x^2+6x+28x+14\).
Question 28
Explanation
Group the first two and last two terms. Their common binomial is \(-2x+3\), leaving \(4x-5\). Expansion verifies \((4x-5)(-2x+3)=-8x^2+12x+10x-15\).
Question 29
Explanation
Group the first two and last two terms. Their common binomial is \(3x-4\), leaving \(5x+2\). Expansion verifies \((5x+2)(3x-4)=15x^2-20x+6x-8\).
Question 30
Explanation
Group the first two and last two terms. Their common binomial is \(2x+6\), leaving \(7x-1\). Expansion verifies \((7x-1)(2x+6)=14x^2+42x-2x-6\).
Question 31
Explanation
Use \(A-B=-(B-A)\) to make the binomial factors identical. Extracting that binomial gives \((x-5)(3x+2)\), which expands to the original.
Question 32
Explanation
Use \(A-B=-(B-A)\) to make the binomial factors identical. Extracting that binomial gives \((y+3)(4y-7)\), which expands to the original.
Question 33
Explanation
Use \(A-B=-(B-A)\) to make the binomial factors identical. Extracting that binomial gives \((a-2)(5a+3)\), which expands to the original.
Question 34
Explanation
Use \(A-B=-(B-A)\) to make the binomial factors identical. Extracting that binomial gives \((m+7)(2m-9)\), which expands to the original.
Question 35
Explanation
Use \(A-B=-(B-A)\) to make the binomial factors identical. Extracting that binomial gives \((p-4)(6p+5)\), which expands to the original.
Question 36
Explanation
Distributing the proposed answer multiplies its outside factor into every inside term and returns \(18x^3-12x^2\). The outside factor is the complete intended GCF.
Question 37
Explanation
Distributing the proposed answer multiplies its outside factor into every inside term and returns \(28y^4+42y^2\). The outside factor is the complete intended GCF.
Question 38
Explanation
Distributing the proposed answer multiplies its outside factor into every inside term and returns \(-32a^5+24a^3\). The outside factor is the complete intended GCF.
Question 39
Explanation
Distributing the proposed answer multiplies its outside factor into every inside term and returns \(45m^4n-60m^2n^3\). The outside factor is the complete intended GCF.
Question 40
Explanation
Distributing the proposed answer multiplies its outside factor into every inside term and returns \(50p^6q^2+75p^3q^5\). The outside factor is the complete intended GCF.
Question 41
Explanation
Reverse distribution gives \(x(7+5)=48\). Substitute the known coefficient sum \(12\), so \(12x=48\) and \(x=4\).
Question 42
Explanation
Reverse distribution gives \(y(9+6)=45\). Substitute the known coefficient sum \(15\), so \(15y=45\) and \(y=3\).
Question 43
Explanation
Reverse distribution gives \(a(11-3)=48\). Substitute the known coefficient sum \(8\), so \(8a=48\) and \(a=6\).
Question 44
Explanation
Reverse distribution gives \(m(4+10)=70\). Substitute the known coefficient sum \(14\), so \(14m=70\) and \(m=5\).
Question 45
Explanation
Reverse distribution gives \(p(13-5)=56\). Substitute the known coefficient sum \(8\), so \(8p=56\) and \(p=7\).
Question 46
Explanation
Reverse distribution and sign normalization show the issue: The inside coefficients still share 2, so the full GCF is 6x^2.
Question 47
Explanation
Reverse distribution and sign normalization show the issue: The inside constant should be -3.
Question 48
Explanation
Reverse distribution and sign normalization show the issue: Extract a negative from one group so both binomials match.
Question 49
Explanation
Reverse distribution and sign normalization show the issue: Expand it and compare every term with the original polynomial.
Question 50
Explanation
Reverse distribution and sign normalization show the issue: This source section is limited to GCF extraction, grouping, and opposite factors.
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Questions to review
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- Question 1Factor a numeric GCFEasy
- Question 2Factor a numeric GCFEasy
- Question 3Factor a numeric GCFEasy
- Question 4Factor a numeric GCFEasy
- Question 5Factor a numeric GCFEasy
- Question 6Factor a variable GCFEasy
- Question 7Factor a variable GCFEasy
- Question 8Factor a variable GCFEasy
- Question 9Factor a variable GCFEasy
- Question 10Factor a variable GCFEasy
- Question 11Factor numeric and variable GCFEasy
- Question 12Factor numeric and variable GCFEasy
- Question 13Factor numeric and variable GCFEasy
- Question 14Factor numeric and variable GCFEasy
- Question 15Factor numeric and variable GCFEasy
- Question 16Factor a negative GCFMedium
- Question 17Factor a negative GCFMedium
- Question 18Factor a negative GCFMedium
- Question 19Factor a negative GCFMedium
- Question 20Factor a negative GCFMedium
- Question 21Factoring by groupingMedium
- Question 22Factoring by groupingMedium
- Question 23Factoring by groupingMedium
- Question 24Factoring by groupingMedium
- Question 25Factoring by groupingMedium
- Question 26Factoring by groupingMedium
- Question 27Factoring by groupingMedium
- Question 28Factoring by groupingMedium
- Question 29Factoring by groupingMedium
- Question 30Factoring by groupingMedium
- Question 31Opposite-factor recognitionMedium
- Question 32Opposite-factor recognitionMedium
- Question 33Opposite-factor recognitionMedium
- Question 34Opposite-factor recognitionMedium
- Question 35Opposite-factor recognitionMedium
- Question 36Equivalent factored expressionMedium
- Question 37Equivalent factored expressionMedium
- Question 38Equivalent factored expressionMedium
- Question 39Equivalent factored expressionMedium
- Question 40Equivalent factored expressionMedium
- Question 41Solve a source-aligned factored relationHard
- Question 42Solve a source-aligned factored relationHard
- Question 43Solve a source-aligned factored relationHard
- Question 44Solve a source-aligned factored relationHard
- Question 45Solve a source-aligned factored relationHard
- Question 46Factoring error analysisHard
- Question 47Factoring error analysisHard
- Question 48Factoring error analysisHard
- Question 49Factoring error analysisHard
- Question 50Factoring error analysisHard