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Algebra 2 Flashcards: Proving And Applying Polynomial Identities

Study Proving And Applying Polynomial Identities in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

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What this deck covers

This deck focuses on Proving And Applying Polynomial Identities, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

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.

Algebra 2 Flashcards: Proving And Applying Polynomial Identities

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QUESTION

What is the factored form of x2−49x^2-49x2−49?

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ANSWER

(x−7)(x+7)(x-7)(x+7)(x−7)(x+7). Recognize 49=7249=7^249=72 and apply difference of squares.

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Flashcard 1: What is the factored form of x2−49x^2-49x2−49?

Answer: (x−7)(x+7)(x-7)(x+7)(x−7)(x+7). Recognize 49=7249=7^249=72 and apply difference of squares.

Flashcard 2: What is the factored form of x3+8x^3+8x3+8?

Answer: (x+2)(x2−2x+4)(x+2)(x^2-2x+4)(x+2)(x2−2x+4). Recognize 8=238=2^38=23 and apply sum of cubes formula.

Flashcard 3: What is the factored form of x3−27x^3-27x3−27?

Answer: (x−3)(x2+3x+9)(x-3)(x^2+3x+9)(x−3)(x2+3x+9). Recognize 27=3327=3^327=33 and apply difference of cubes formula.

Flashcard 4: What is the expanded form of (x+y)2−(x−y)2(x+y)^2-(x-y)^2(x+y)2−(x−y)2?

Answer: 4xy4xy4xy. Difference of squares: (x+y)2−(x−y)2=4xy(x+y)^2-(x-y)^2 = 4xy(x+y)2−(x−y)2=4xy.

Flashcard 5: What is the factored form of x2−2xy+y2x^2-2xy+y^2x2−2xy+y2?

Answer: (x−y)2(x-y)^2(x−y)2. Perfect square trinomial with negative middle term.

Flashcard 6: What is the factored form of x2+2xy+y2x^2+2xy+y^2x2+2xy+y2?

Answer: (x+y)2(x+y)^2(x+y)2. Perfect square trinomial with all positive terms.

Flashcard 7: What condition on mmm and nnn guarantees a=m2−n2a=m^2-n^2a=m2−n2 is positive?

Answer: m>nm>nm>n. Ensures the first leg aaa is positive in the formula.

Flashcard 8: What is a2+b2a^2+b^2a2+b2 if a=m2−n2a=m^2-n^2a=m2−n2 and b=2mnb=2mnb=2mn?

Answer: (m2+n2)2(m^2+n^2)^2(m2+n2)2. The hypotenuse squared in Pythagorean triple formula.

Flashcard 9: What is (m2−n2)2+(2mn)2(m^2-n^2)^2+(2mn)^2(m2−n2)2+(2mn)2 simplified to one expression?

Answer: (m2+n2)2(m^2+n^2)^2(m2+n2)2. Direct application of the Pythagorean identity (x2+y2)2(x^2+y^2)^2(x2+y2)2.

Flashcard 10: What identity verifies (m2−n2)2+(2mn)2=(m2+n2)2(m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2(m2−n2)2+(2mn)2=(m2+n2)2?

Answer: (x2+y2)2=(x2−y2)2+(2xy)2(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2(x2+y2)2=(x2−y2)2+(2xy)2. The identity that validates all Pythagorean triple formulas.

Flashcard 11: What triple results from m=4m=4m=4 and n=3n=3n=3 using a=m2−n2a=m^2-n^2a=m2−n2, b=2mnb=2mnb=2mn, c=m2+n2c=m^2+n^2c=m2+n2?

Answer: (7,24,25)(7,24,25)(7,24,25). Calculate: a=16−9=7a=16-9=7a=16−9=7, b=24b=24b=24, c=16+9=25c=16+9=25c=16+9=25.

Flashcard 12: What triple results from m=4m=4m=4 and n=1n=1n=1 using a=m2−n2a=m^2-n^2a=m2−n2, b=2mnb=2mnb=2mn, c=m2+n2c=m^2+n^2c=m2+n2?

Answer: (15,8,17)(15,8,17)(15,8,17). Calculate: a=16−1=15a=16-1=15a=16−1=15, b=8b=8b=8, c=16+1=17c=16+1=17c=16+1=17.

Flashcard 13: What triple results from m=3m=3m=3 and n=2n=2n=2 using a=m2−n2a=m^2-n^2a=m2−n2, b=2mnb=2mnb=2mn, c=m2+n2c=m^2+n^2c=m2+n2?

Answer: (5,12,13)(5,12,13)(5,12,13). Calculate: a=9−4=5a=9-4=5a=9−4=5, b=12b=12b=12, c=9+4=13c=9+4=13c=9+4=13.

Flashcard 14: What triple results from m=2m=2m=2 and n=1n=1n=1 using a=m2−n2a=m^2-n^2a=m2−n2, b=2mnb=2mnb=2mn, c=m2+n2c=m^2+n^2c=m2+n2?

Answer: (3,4,5)(3,4,5)(3,4,5). Calculate: a=4−1=3a=4-1=3a=4−1=3, b=4b=4b=4, c=4+1=5c=4+1=5c=4+1=5.

Flashcard 15: What is the expanded form of (x2−3x)2(x^2-3x)^2(x2−3x)2?

Answer: x4−6x3+9x2x^4-6x^3+9x^2x4−6x3+9x2. Square the binomial x2−3xx^2-3xx2−3x using (a−b)2(a-b)^2(a−b)2 pattern.

Flashcard 16: What is the expanded form of (2xy)2(2xy)^2(2xy)2?

Answer: 4x2y24x^2y^24x2y2. Square each factor: (2xy)2=22⋅x2⋅y2(2xy)^2 = 2^2 \cdot x^2 \cdot y^2(2xy)2=22⋅x2⋅y2.

Flashcard 17: What simplified expression equals (x2−y2)2+(2xy)2(x^2-y^2)^2+(2xy)^2(x2−y2)2+(2xy)2?

Answer: x4+2x2y2+y4x^4+2x^2y^2+y^4x4+2x2y2+y4. Adding the two expressions equals (x2+y2)2(x^2+y^2)^2(x2+y2)2.

Flashcard 18: What identity results from subtracting (a−b)2(a-b)^2(a−b)2 from (a+b)2(a+b)^2(a+b)2?

Answer: (a+b)2−(a−b)2=4ab(a+b)^2-(a-b)^2=4ab(a+b)2−(a−b)2=4ab. Difference of perfect squares yields four times the product.

Flashcard 19: What identity results from adding (a+b)2(a+b)^2(a+b)2 and (a−b)2(a-b)^2(a−b)2?

Answer: (a+b)2+(a−b)2=2(a2+b2)(a+b)^2+(a-b)^2=2(a^2+b^2)(a+b)2+(a−b)2=2(a2+b2). Sum of perfect squares yields twice the sum of squares.

Flashcard 20: What is the expanded form of (x+3)2(x+3)^2(x+3)2?

Answer: x2+6x+9x^2+6x+9x2+6x+9. Apply (a+b)2(a+b)^2(a+b)2 with a=xa=xa=x and b=3b=3b=3.

Flashcard 21: What is the expanded form of (2x−5)2(2x-5)^2(2x−5)2?

Answer: 4x2−20x+254x^2-20x+254x2−20x+25. Apply (a−b)2(a-b)^2(a−b)2 with a=2xa=2xa=2x and b=5b=5b=5.

Flashcard 22: What is the expanded form of (x−4)(x+4)(x-4)(x+4)(x−4)(x+4)?

Answer: x2−16x^2-16x2−16. Apply difference of squares: a2−b2a^2-b^2a2−b2 with a=xa=xa=x, b=4b=4b=4.

Flashcard 23: What is the expanded form of (x+2)3(x+2)^3(x+2)3?

Answer: x3+6x2+12x+8x^3+6x^2+12x+8x3+6x2+12x+8. Apply (a+b)3(a+b)^3(a+b)3 formula with a=xa=xa=x and b=2b=2b=2.

Flashcard 24: What is the identity for (a−b)(a2+ab+b2)(a-b)(a^2+ab+b^2)(a−b)(a2+ab+b2)?

Answer: (a−b)(a2+ab+b2)=a3−b3(a-b)(a^2+ab+b^2)=a^3-b^3(a−b)(a2+ab+b2)=a3−b3. Difference of cubes factorization produces the cubic expansion.

Flashcard 25: What is the identity for (a+b)(a2−ab+b2)(a+b)(a^2-ab+b^2)(a+b)(a2−ab+b2)?

Answer: (a+b)(a2−ab+b2)=a3+b3(a+b)(a^2-ab+b^2)=a^3+b^3(a+b)(a2−ab+b2)=a3+b3. Sum of cubes factorization produces the cubic expansion.

Flashcard 26: What is the expanded form of (x+y)2+(x−y)2(x+y)^2+(x-y)^2(x+y)2+(x−y)2?

Answer: 2x2+2y22x^2+2y^22x2+2y2. Sum of squares: (x+y)2+(x−y)2=2(x2+y2)(x+y)^2+(x-y)^2 = 2(x^2+y^2)(x+y)2+(x−y)2=2(x2+y2).

Flashcard 27: What is the Pythagorean-triple formula from mmm and nnn: a,b,ca,b,ca,b,c?

Answer: a=m2−n2,  b=2mn,  c=m2+n2a=m^2-n^2,\;b=2mn,\;c=m^2+n^2a=m2−n2,b=2mn,c=m2+n2. Standard parametric form generating all primitive Pythagorean triples.

Flashcard 28: What perfect-square trinomial equals x2−12x+36x^2-12x+36x2−12x+36?

Answer: (x−6)2(x-6)^2(x−6)2. Recognize pattern: 2cdot6=122 cdot 6 = 122cdot6=12 confirms perfect square.

Flashcard 29: What perfect-square trinomial equals x2+10x+25x^2+10x+25x2+10x+25?

Answer: (x+5)2(x+5)^2(x+5)2. Recognize pattern: 2cdot5=102 cdot 5 = 102cdot5=10 confirms perfect square.

Flashcard 30: What is the expanded form of (5x−1)2(5x-1)^2(5x−1)2?

Answer: 25x2−10x+125x^2-10x+125x2−10x+1. Apply formula with a=5a=5a=5, b=−1b=-1b=−1.