Algebra Flashcards: Using Structure To Rewrite Expressions

What this deck covers

This deck focuses on Using Structure To Rewrite Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Factor completely using structure: $x^6-y^6$.

Answer: $(x^3-y^3)(x^3+y^3)$. Sees $x^6-y^6$ as $(x^3)^2-(y^3)^2$, a difference of squares.

Flashcard 2: Factor by recognizing a difference of squares: $x^2-64$.

Answer: $(x-8)(x+8)$. Recognizes $x^2-8^2$ as a difference of squares.

Flashcard 3: Factor completely by recognizing a perfect square: $16p^2+24p+9$.

Answer: $(4p+3)^2$. Perfect square trinomial with $a=4p$, $b=3$.

Flashcard 4: What is the greatest common factor rewrite for $ax+ay$?

Answer: $ax+ay=a(x+y)$. Factors out the common factor $a$.

Flashcard 5: Factor using sum of cubes: $8y^3+125$.

Answer: $(2y+5)(4y^2-10y+25)$. Recognizes $(2y)^3+5^3$ as a sum of cubes.

Flashcard 6: Rewrite by factoring out the greatest common factor: $-10x^3+5x^2$.

Answer: $5x^2(1-2x)$. Factor out $5x^2$ from both terms.

Flashcard 7: Factor completely using structure: $x^2+14x+49$.

Answer: $(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.

Flashcard 8: Rewrite by factoring out the greatest common factor: $15y^2+20y$.

Answer: $5y(3y+4)$. Greatest common factor is $5y$.

Flashcard 9: What is the factoring pattern for a difference of cubes $a^3-b^3$?

Answer: $a^3-b^3=(a-b)(a^2+ab+b^2)$. Uses the difference of cubes factoring formula.

Flashcard 10: Rewrite by factoring out the greatest common factor: $15y^2+20y$.

Answer: $5y(3y+4)$. Greatest common factor is $5y$.

Flashcard 11: Factor using sum of cubes: $x^3+27$.

Answer: $(x+3)(x^2-3x+9)$. Recognizes $x^3+3^3$ as a sum of cubes.

Flashcard 12: What is the factoring pattern for a sum of cubes $a^3+b^3$?

Answer: $a^3+b^3=(a+b)(a^2-ab+b^2)$. Uses the sum of cubes factoring formula.

Flashcard 13: Factor completely by recognizing a perfect square: $25m^2-30m+9$.

Answer: $(5m-3)^2$. Perfect square trinomial with $a=5m$, $b=3$.

Flashcard 14: Rewrite by factoring out a negative: $-4x^2+12x-9$.

Answer: $-(2x-3)^2$. Factors out $-1$ to make leading coefficient positive.

Flashcard 15: Factor completely by using structure: $x^4-1$.

Answer: $(x-1)(x+1)(x^2+1)$. Factors $(x^2)^2-1^2$ then continues factoring $x^2-1$.

Flashcard 16: Rewrite by using structure to factor: $x^2-(y+1)^2$.

Answer: $(x-(y+1))(x+(y+1))$. Treats $(y+1)$ as a single term in difference of squares.

Flashcard 17: What is the factoring pattern for a perfect square trinomial $a^2-2ab+b^2$?

Answer: $a^2-2ab+b^2=(a-b)^2$. Square of a binomial difference pattern.

Flashcard 18: What is the exponent rule rewrite for $x^m\cdot x^n$?

Answer: $x^m\cdot x^n=x^{m+n}$. Add exponents when multiplying same bases.

Flashcard 19: Rewrite by using exponent structure: $(x^2)^3$.

Answer: $x^6$. Power of a power rule: $(x^2)^3=x^{2\cdot 3}$.

Flashcard 20: Rewrite by using exponent structure: $(2x^3)^2$.

Answer: $4x^6$. Power of a product: $(2x^3)^2=2^2\cdot (x^3)^2$.

Flashcard 21: Factor by recognizing a difference of squares: $100-t^2$.

Answer: $(10-t)(10+t)$. Recognizes $10^2-t^2$ as a difference of squares.

Flashcard 22: Identify the structure of $x^2+10x+25$ as a single squared binomial.

Answer: $(x+5)^2$. Form $a^2+2ab+b^2$ with $a=x$, $b=5$.

Flashcard 23: Factor using difference of cubes: $64a^3-1$.

Answer: $(4a-1)(16a^2+4a+1)$. Recognizes $(4a)^3-1^3$ as a difference of cubes.

Flashcard 24: Factor completely using repeated difference of squares: $x^4-16$.

Answer: $(x-2)(x+2)(x^2+4)$. Factor $(x^2-4)$ further as $(x-2)(x+2)$.

Flashcard 25: Rewrite by factoring out the greatest common factor: $8a^3-12a^2$.

Answer: $4a^2(2a-3)$. Greatest common factor is $4a^2$.

Flashcard 26: Factor by grouping using structure: $3x^2+6x+2x+4$.

Answer: $(3x+2)(x+2)$. Groups $3x^2+6x$ and $2x+4$ with common factors.

Flashcard 27: What is the factoring pattern for a difference of squares $a^2-b^2$?

Answer: $a^2-b^2=(a-b)(a+b)$. Product of sum and difference of the same terms.

Flashcard 28: Rewrite by factoring using a common binomial: $x(x-3)+5(x-3)$.

Answer: $(x-3)(x+5)$. Factor out the common binomial $(x-3)$.

Flashcard 29: Rewrite by using exponent structure: $x^2\cdot x^5$.

Answer: $x^7$. Product rule for exponents: $x^2\cdot x^5=x^{2+5}$.

Flashcard 30: Factor completely using repeated difference of squares: $x^4-16$.

Answer: $(x-2)(x+2)(x^2+4)$. Factor $(x^2-4)$ further as $(x-2)(x+2)$.

Flashcard 31: Rewrite by using structure to factor: $9-(x-2)^2$.

Answer: $(5-x)(x+1)$. Recognizes $3^2-(x-2)^2$ as a difference of squares.

Flashcard 32: Factor completely using structure: $x^2+14x+49$.

Answer: $(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.

Flashcard 33: Rewrite by using structure to factor: $x^2-(y+1)^2$.

Answer: $(x-(y+1))(x+(y+1))$. Treats $(y+1)$ as a single term in difference of squares.

Flashcard 34: Factor completely by using structure: $x^4-2x^2+1$.

Answer: $(x^2-1)^2$. Sees structure as $(x^2-1)^2$ where $u=x^2-1$.

Flashcard 35: What is the factoring pattern for a sum of cubes $a^3+b^3$?

Answer: $a^3+b^3=(a+b)(a^2-ab+b^2)$. Uses the sum of cubes factoring formula.

Flashcard 36: Factor completely using structure: $x^2-12x+36$.

Answer: $(x-6)^2$. Perfect square trinomial with $a=x$, $b=6$.

Flashcard 37: What is the exponent rule rewrite for $(x^m)^n$?

Answer: $(x^m)^n=x^{mn}$. Multiply exponents when raising a power to a power.

Flashcard 38: Rewrite by using exponent structure: $x^2\cdot x^5$.

Answer: $x^7$. Product rule for exponents: $x^2\cdot x^5=x^{2+5}$.

Flashcard 39: What is the exponent rule rewrite for $x^m\div x^n$ (with $x\neq 0$)?

Answer: $x^m\div x^n=x^{m-n}$. Subtract exponents when dividing same bases.

Flashcard 40: Factor completely using structure: $x^6-y^6$.

Answer: $(x^3-y^3)(x^3+y^3)$. Sees $x^6-y^6$ as $(x^3)^2-(y^3)^2$, a difference of squares.

Flashcard 41: Factor using sum of cubes: $x^3+27$.

Answer: $(x+3)(x^2-3x+9)$. Recognizes $x^3+3^3$ as a sum of cubes.

Flashcard 42: Rewrite by factoring out the greatest common factor: $6x^2+9x$.

Answer: $3x(2x+3)$. Greatest common factor is $3x$.

Flashcard 43: Rewrite by factoring using a common binomial: $(x+2)(x-1)-(x+2)3$.

Answer: $(x+2)(x-4)$. Factor out the common binomial $(x+2)$.

Flashcard 44: Factor completely by recognizing a difference of squares: $49y^2-1$.

Answer: $(7y-1)(7y+1)$. Recognizes $(7y)^2-1^2$ as a difference of squares.

Flashcard 45: Factor by grouping using structure: $ax+ay+bx+by$.

Answer: $(a+b)(x+y)$. Group terms with common factors $(a+b)$ and $(x+y)$.

Flashcard 46: Identify the structure of $x^2-10x+25$ as a single squared binomial.

Answer: $(x-5)^2$. Form $a^2-2ab+b^2$ with $a=x$, $b=5$.

Flashcard 47: Rewrite by factoring out the greatest common factor: $8a^3-12a^2$.

Answer: $4a^2(2a-3)$. Greatest common factor is $4a^2$.

Flashcard 48: Factor completely by using structure: $x^4-16$.

Answer: $(x^2-4)(x^2+4)$. Sees $x^4-16$ as $(x^2)^2-4^2$, a difference of squares.

Flashcard 49: Rewrite by factoring out a negative to make the leading term positive: $-x^2+6x-9$.

Answer: $-(x-3)^2$. Factors out $-1$ to make leading coefficient positive.

Flashcard 50: Factor by grouping using structure: $ax+ay+bx+by$.

Answer: $(a+b)(x+y)$. Group terms with common factors $(a+b)$ and $(x+y)$.

Flashcard 51: Rewrite by using exponent structure: $(2x^3)^2$.

Answer: $4x^6$. Power of a product: $(2x^3)^2=2^2\cdot (x^3)^2$.

Flashcard 52: Identify the structure of $x^2-10x+25$ as a single squared binomial.

Answer: $(x-5)^2$. Form $a^2-2ab+b^2$ with $a=x$, $b=5$.

Flashcard 53: Factor using difference of cubes: $x^3-8$.

Answer: $(x-2)(x^2+2x+4)$. Recognizes $x^3-2^3$ as a difference of cubes.

Flashcard 54: Factor by grouping using structure: $x^2+5x+2x+10$.

Answer: $(x+5)(x+2)$. Groups $x^2+5x$ and $2x+10$ with common factors.

Flashcard 55: What is the factoring pattern for a perfect square trinomial $a^2+2ab+b^2$?

Answer: $a^2+2ab+b^2=(a+b)^2$. Square of a binomial sum pattern.

Flashcard 56: Factor by recognizing a difference of squares: $100-t^2$.

Answer: $(10-t)(10+t)$. Recognizes $10^2-t^2$ as a difference of squares.

Flashcard 57: Rewrite by using exponent structure: $(x^2)^3$.

Answer: $x^6$. Power of a power rule: $(x^2)^3=x^{2\cdot 3}$.

Flashcard 58: Rewrite by factoring out the greatest common factor: $-10x^3+5x^2$.

Answer: $5x^2(1-2x)$. Factor out $5x^2$ from both terms.

Flashcard 59: Factor completely by using structure: $x^4-1$.

Answer: $(x-1)(x+1)(x^2+1)$. Factors $(x^2)^2-1^2$ then continues factoring $x^2-1$.

Flashcard 60: What is the factoring pattern for a perfect square trinomial $a^2-2ab+b^2$?

Answer: $a^2-2ab+b^2=(a-b)^2$. Square of a binomial difference pattern.

Flashcard 61: Factor by grouping using structure: $3x^2+6x+2x+4$.

Answer: $(3x+2)(x+2)$. Groups $3x^2+6x$ and $2x+4$ with common factors.

Flashcard 62: Factor completely using structure: $x^2-12x+36$.

Answer: $(x-6)^2$. Perfect square trinomial with $a=x$, $b=6$.

Flashcard 63: Factor using difference of cubes: $64a^3-1$.

Answer: $(4a-1)(16a^2+4a+1)$. Recognizes $(4a)^3-1^3$ as a difference of cubes.

Flashcard 64: What is the factoring pattern for a difference of cubes $a^3-b^3$?

Answer: $a^3-b^3=(a-b)(a^2+ab+b^2)$. Uses the difference of cubes factoring formula.

Flashcard 65: Rewrite by using structure to factor: $(x+4)^2-9$.

Answer: $(x+1)(x+7)$. Recognizes $(x+4)^2-3^2$ as a difference of squares.

Flashcard 66: Rewrite by using structure to factor: $(x+4)^2-9$.

Answer: $(x+1)(x+7)$. Recognizes $(x+4)^2-3^2$ as a difference of squares.

Flashcard 67: What is the distributive property rewrite for $a(b+c)$?

Answer: $a(b+c)=ab+ac$. Distributes $a$ to both terms in parentheses.

Flashcard 68: Factor by grouping using structure: $2x^2-8x+3x-12$.

Answer: $(2x+3)(x-4)$. Groups $2x^2-8x$ and $3x-12$ with common factors.

Flashcard 69: Identify the structure and factor completely: $x^4-y^4$.

Answer: $(x^2-y^2)(x^2+y^2)$. Recognizes $x^4-y^4$ as $(x^2)^2-(y^2)^2$, a difference of squares.

Flashcard 70: Factor by recognizing a difference of squares: $x^2-64$.

Answer: $(x-8)(x+8)$. Recognizes $x^2-8^2$ as a difference of squares.

Flashcard 71: Factor completely by recognizing a difference of squares: $9x^2-25$.

Answer: $(3x-5)(3x+5)$. Recognizes $(3x)^2-5^2$ as a difference of squares.

Flashcard 72: What is the exponent rule rewrite for $x^m\div x^n$ (with $x\neq 0$)?

Answer: $x^m\div x^n=x^{m-n}$. Subtract exponents when dividing same bases.

Flashcard 73: Factor using difference of cubes: $x^3-8$.

Answer: $(x-2)(x^2+2x+4)$. Recognizes $x^3-2^3$ as a difference of cubes.

Flashcard 74: Identify the common binomial factor in $x(x-3)+5(x-3)$.

Answer: $(x-3)$. Both terms contain the binomial factor $(x-3)$.

Flashcard 75: What is the exponent rule rewrite for $(x^m)^n$?

Answer: $(x^m)^n=x^{mn}$. Multiply exponents when raising a power to a power.

Flashcard 76: Factor completely by recognizing a difference of squares: $4a^2-81$.

Answer: $(2a-9)(2a+9)$. Recognizes $(2a)^2-9^2$ as a difference of squares.

Flashcard 77: Factor by grouping using structure: $2x^2-8x+3x-12$.

Answer: $(2x+3)(x-4)$. Groups $2x^2-8x$ and $3x-12$ with common factors.

Flashcard 78: Factor completely by recognizing a difference of squares: $49y^2-1$.

Answer: $(7y-1)(7y+1)$. Recognizes $(7y)^2-1^2$ as a difference of squares.

Flashcard 79: Identify the common binomial factor in $x(x-3)+5(x-3)$.

Answer: $(x-3)$. Both terms contain the binomial factor $(x-3)$.

Flashcard 80: What is the factoring pattern for a perfect square trinomial $a^2+2ab+b^2$?

Answer: $a^2+2ab+b^2=(a+b)^2$. Square of a binomial sum pattern.

Flashcard 81: Rewrite by using structure to factor: $(2x-1)^2-16$.

Answer: $(2x-5)(2x+3)$. Recognizes $(2x-1)^2-4^2$ as a difference of squares.

Flashcard 82: Identify the structure and factor completely: $x^4-y^4$.

Answer: $(x^2-y^2)(x^2+y^2)$. Recognizes $x^4-y^4$ as $(x^2)^2-(y^2)^2$, a difference of squares.

Flashcard 83: What is the exponent rule rewrite for $x^m\cdot x^n$?

Answer: $x^m\cdot x^n=x^{m+n}$. Add exponents when multiplying same bases.

Flashcard 84: Rewrite by factoring out a negative: $-4x^2+12x-9$.

Answer: $-(2x-3)^2$. Factors out $-1$ to make leading coefficient positive.

Flashcard 85: Factor completely by recognizing a perfect square: $25m^2-30m+9$.

Answer: $(5m-3)^2$. Perfect square trinomial with $a=5m$, $b=3$.

Flashcard 86: Factor completely by using structure: $x^4-16$.

Answer: $(x^2-4)(x^2+4)$. Sees $x^4-16$ as $(x^2)^2-4^2$, a difference of squares.

Flashcard 87: Factor completely by recognizing a perfect square: $16p^2+24p+9$.

Answer: $(4p+3)^2$. Perfect square trinomial with $a=4p$, $b=3$.

Flashcard 88: Factor completely by recognizing a difference of squares: $9x^2-25$.

Answer: $(3x-5)(3x+5)$. Recognizes $(3x)^2-5^2$ as a difference of squares.

Flashcard 89: Factor completely by recognizing a difference of squares: $4a^2-81$.

Answer: $(2a-9)(2a+9)$. Recognizes $(2a)^2-9^2$ as a difference of squares.

Flashcard 90: Rewrite by using structure to factor: $(2x-1)^2-16$.

Answer: $(2x-5)(2x+3)$. Recognizes $(2x-1)^2-4^2$ as a difference of squares.

Flashcard 91: Factor using sum of cubes: $8y^3+125$.

Answer: $(2y+5)(4y^2-10y+25)$. Recognizes $(2y)^3+5^3$ as a sum of cubes.

Flashcard 92: Identify the structure of $x^2+10x+25$ as a single squared binomial.

Answer: $(x+5)^2$. Form $a^2+2ab+b^2$ with $a=x$, $b=5$.

Flashcard 93: Factor completely by using structure: $x^4-2x^2+1$.

Answer: $(x^2-1)^2$. Sees structure as $(x^2-1)^2$ where $u=x^2-1$.

Flashcard 94: Rewrite by using structure to factor: $9-(x-2)^2$.

Answer: $(5-x)(x+1)$. Recognizes $3^2-(x-2)^2$ as a difference of squares.

Flashcard 95: Rewrite by using exponent structure: $\frac{x^9}{x^4}$ (with $x\neq 0$).

Answer: $x^5$. Quotient rule for exponents: $x^9\div x^4=x^{9-4}$.

Flashcard 96: Rewrite by using exponent structure: $\frac{x^9}{x^4}$ (with $x\neq 0$).

Answer: $x^5$. Quotient rule for exponents: $x^9\div x^4=x^{9-4}$.

Flashcard 97: Factor by grouping using structure: $x^2+5x+2x+10$.

Answer: $(x+5)(x+2)$. Groups $x^2+5x$ and $2x+10$ with common factors.

Flashcard 98: Rewrite by factoring out the greatest common factor: $6x^2+9x$.

Answer: $3x(2x+3)$. Greatest common factor is $3x$.

Flashcard 99: What is the distributive property rewrite for $a(b+c)$?

Answer: $a(b+c)=ab+ac$. Distributes $a$ to both terms in parentheses.

Flashcard 100: Rewrite by factoring out a negative to make the leading term positive: $-x^2+6x-9$.

Answer: $-(x-3)^2$. Factors out $-1$ to make leading coefficient positive.