Practice
FOIL Method and Special Products Practice
Fifty original questions on distribution, binomial squares, conjugates, recognition, equivalence, and error analysis.
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Question 1
Explanation
The four products are \(8x^2\), \(-10x\), \(12x\), and \(-15\). Combining the middle terms gives \(8x^{2}+2x-15\).
Question 2
Explanation
The four products are \(6x^2\), \(21x\), \(-4x\), and \(-14\). Combining the middle terms gives \(6x^{2}+17x-14\).
Question 3
Explanation
The four products are \(-6x^2\), \(-8x\), \(15x\), and \(20\). Combining the middle terms gives \(-6x^{2}+7x+20\).
Question 4
Explanation
The four products are \(-12x^2\), \(8x\), \(-3x\), and \(2\). Combining the middle terms gives \(-12x^{2}+5x+2\).
Question 5
Explanation
The four products are \(10x^2\), \(-5x\), \(-12x\), and \(6\). Combining the middle terms gives \(10x^{2}-17x+6\).
Question 6
Explanation
Use \((A+B)^2=A^2+2AB+B^2\) with \(A=1x\) and \(B=4\). The result is \(x^{2}+8x+16\).
Question 7
Explanation
Use \((A+B)^2=A^2+2AB+B^2\) with \(A=2x\) and \(B=3\). The result is \(4x^{2}+12x+9\).
Question 8
Explanation
Use \((A+B)^2=A^2+2AB+B^2\) with \(A=3x\) and \(B=5\). The result is \(9x^{2}+30x+25\).
Question 9
Explanation
Use \((A+B)^2=A^2+2AB+B^2\) with \(A=4x\) and \(B=2\). The result is \(16x^{2}+16x+4\).
Question 10
Explanation
Use \((A+B)^2=A^2+2AB+B^2\) with \(A=5x\) and \(B=7\). The result is \(25x^{2}+70x+49\).
Question 11
Explanation
Use \((A-B)^2=A^2-2AB+B^2\). The last term remains positive, so the expansion is \(x^{2}-12x+36\).
Question 12
Explanation
Use \((A-B)^2=A^2-2AB+B^2\). The last term remains positive, so the expansion is \(4x^{2}-20x+25\).
Question 13
Explanation
Use \((A-B)^2=A^2-2AB+B^2\). The last term remains positive, so the expansion is \(9x^{2}-12x+4\).
Question 14
Explanation
Use \((A-B)^2=A^2-2AB+B^2\). The last term remains positive, so the expansion is \(16x^{2}-56x+49\).
Question 15
Explanation
Use \((A-B)^2=A^2-2AB+B^2\). The last term remains positive, so the expansion is \(25x^{2}-30x+9\).
Question 16
Explanation
The factors are conjugates, so the outer and inner terms cancel. The difference of squares is \(x^{2}-64\).
Question 17
Explanation
The factors are conjugates, so the outer and inner terms cancel. The difference of squares is \(4x^{2}-9\).
Question 18
Explanation
The factors are conjugates, so the outer and inner terms cancel. The difference of squares is \(9x^{2}-49\).
Question 19
Explanation
The factors are conjugates, so the outer and inner terms cancel. The difference of squares is \(16x^{2}-25\).
Question 20
Explanation
The factors are conjugates, so the outer and inner terms cancel. The difference of squares is \(36x^{2}-4\).
Question 21
Explanation
The middle coefficient is the sum \(2+3=5\), and the constant is the product \(2(3)=6\). Thus the expansion is \(m^2+5m+6\).
Question 22
Explanation
The middle coefficient is the sum \(4+-5=-1\), and the constant is the product \(4(-5)=-20\). Thus the expansion is \(n^2-1n-20\).
Question 23
Explanation
The middle coefficient is the sum \(-3+2=-1\), and the constant is the product \(-3(2)=-6\). Thus the expansion is \(p^2-1p-6\).
Question 24
Explanation
The middle coefficient is the sum \(5+6=11\), and the constant is the product \(5(6)=30\). Thus the expansion is \(q^2+11q+30\).
Question 25
Explanation
The middle coefficient is the sum \(-2+-7=-9\), and the constant is the product \(-2(-7)=14\). Thus the expansion is \(r^2-9r+14\).
Question 26
Explanation
The end terms and middle-term sign match the square of a sum identity. Expanding \((x+7)^2\) reproduces \(x^2+14x+49\).
Question 27
Explanation
The end terms and middle-term sign match the square of a difference identity. Expanding \((2x-5)^2\) reproduces \(4x^2-20x+25\).
Question 28
Explanation
The end terms and middle-term sign match the difference of squares identity. Expanding \((3y+8)(3y-8)\) reproduces \(9y^2-64\).
Question 29
Explanation
The end terms and middle-term sign match the square of a sum identity. Expanding \((5p+3)^2\) reproduces \(25p^2+30p+9\).
Question 30
Explanation
The end terms and middle-term sign match the square of a difference identity. Expanding \((4q-3)^2\) reproduces \(16q^2-24q+9\).
Question 31
Explanation
Symbolic multiplication gives coefficient vector \([4,11,6]\), so the equivalent polynomial is \(6x^{2}+11x+4\).
Question 32
Explanation
Symbolic multiplication gives coefficient vector \([-10,1,3]\), so the equivalent polynomial is \(3x^{2}+x-10\).
Question 33
Explanation
Symbolic multiplication gives coefficient vector \([-3,2,8]\), so the equivalent polynomial is \(8x^{2}+2x-3\).
Question 34
Explanation
Symbolic multiplication gives coefficient vector \([8,-26,15]\), so the equivalent polynomial is \(15x^{2}-26x+8\).
Question 35
Explanation
Symbolic multiplication gives coefficient vector \([-7,23,-6]\), so the equivalent polynomial is \(-6x^{2}+23x-7\).
Question 36
Explanation
Both end terms are squares, and the middle term equals twice their signed product. Therefore \(x^2+18x+81=(x+9)^2\).
Question 37
Explanation
Both end terms are squares, and the middle term equals twice their signed product. Therefore \(4x^2-12x+9=(2x-3)^2\).
Question 38
Explanation
Both end terms are squares, and the middle term equals twice their signed product. Therefore \(9x^2+42x+49=(3x+7)^2\).
Question 39
Explanation
Both end terms are squares, and the middle term equals twice their signed product. Therefore \(25x^2-40x+16=(5x-4)^2\).
Question 40
Explanation
Both end terms are squares, and the middle term equals twice their signed product. Therefore \(36x^2+60x+25=(6x+5)^2\).
Question 41
Explanation
The complete distributive expansion shows: The middle term 12x. Tracking first, outer, inner, and last products prevents the stated error.
Question 42
Explanation
The complete distributive expansion shows: The square needs the middle term -20x and a positive constant 25. Tracking first, outer, inner, and last products prevents the stated error.
Question 43
Explanation
The complete distributive expansion shows: The constant must be -16. Tracking first, outer, inner, and last products prevents the stated error.
Question 44
Explanation
The complete distributive expansion shows: The second factor has three terms, so complete distribution requires six products before combining. Tracking first, outer, inner, and last products prevents the stated error.
Question 45
Explanation
The complete distributive expansion shows: 19, because 15x+4x=19x. Tracking first, outer, inner, and last products prevents the stated error.
Question 46
Explanation
The binomial product is \(4x^{2}-9\). Distributing \(x+1\) across that result and combining like terms gives \(4x^{3}+4x^{2}-9x-9\).
Question 47
Explanation
The binomial product is \(9x^{2}-16\). Distributing \(x+1\) across that result and combining like terms gives \(9x^{3}+9x^{2}-16x-16\).
Question 48
Explanation
The binomial product is \(16x^{2}-25\). Distributing \(x+1\) across that result and combining like terms gives \(16x^{3}+16x^{2}-25x-25\).
Question 49
Explanation
The binomial product is \(25x^{2}-4\). Distributing \(x+1\) across that result and combining like terms gives \(25x^{3}+25x^{2}-4x-4\).
Question 50
Explanation
The binomial product is \(36x^{2}-1\). Distributing \(x+1\) across that result and combining like terms gives \(36x^{3}+36x^{2}-x-1\).
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Questions to review
No mistakes this time. Excellent work.
- Question 1FOILEasy
- Question 2FOILEasy
- Question 3FOILEasy
- Question 4FOILEasy
- Question 5FOILEasy
- Question 6Square of a sumEasy
- Question 7Square of a sumEasy
- Question 8Square of a sumEasy
- Question 9Square of a sumEasy
- Question 10Square of a sumEasy
- Question 11Square of a differenceEasy
- Question 12Square of a differenceEasy
- Question 13Square of a differenceEasy
- Question 14Square of a differenceEasy
- Question 15Square of a differenceEasy
- Question 16Product of sum and differenceMedium
- Question 17Product of sum and differenceMedium
- Question 18Product of sum and differenceMedium
- Question 19Product of sum and differenceMedium
- Question 20Product of sum and differenceMedium
- Question 21Symbolic binomial multiplicationMedium
- Question 22Symbolic binomial multiplicationMedium
- Question 23Symbolic binomial multiplicationMedium
- Question 24Symbolic binomial multiplicationMedium
- Question 25Symbolic binomial multiplicationMedium
- Question 26Special-product recognitionMedium
- Question 27Special-product recognitionMedium
- Question 28Special-product recognitionMedium
- Question 29Special-product recognitionMedium
- Question 30Special-product recognitionMedium
- Question 31Equivalent-expression expansionMedium
- Question 32Equivalent-expression expansionMedium
- Question 33Equivalent-expression expansionMedium
- Question 34Equivalent-expression expansionMedium
- Question 35Equivalent-expression expansionMedium
- Question 36Reverse special-product recognitionMedium
- Question 37Reverse special-product recognitionMedium
- Question 38Reverse special-product recognitionMedium
- Question 39Reverse special-product recognitionMedium
- Question 40Reverse special-product recognitionMedium
- Question 41Square of a sum errorHard
- Question 42Square of a difference errorHard
- Question 43Conjugate errorHard
- Question 44FOIL scopeHard
- Question 45FOIL verificationHard
- Question 46Multi-step polynomial multiplicationHard
- Question 47Multi-step polynomial multiplicationHard
- Question 48Multi-step polynomial multiplicationHard
- Question 49Multi-step polynomial multiplicationHard
- Question 50Multi-step polynomial multiplicationHard