Using Structure to Rewrite Expressions - Algebra 1

The expression $x^2 + 6x + 9 - y^2$ can be restructured to reveal which factoring pattern?
Answer: A difference of squares after recognizing $(x + 3)^2 - y^2$ The correct answer is B. Recognizing that $x^2 + 6x + 9 = (x + 3)^2$ , the expression becomes $(x + 3)^2 - y^2$ , which is a difference of squares. This factors as $[(x + 3) - y][(x + 3) + y] = (x + 3 - y)(x + 3 + y)$ . Choice A doesn't identify the specific pattern correctly. Choice C is wrong because this expression can be factored and is not a sum of squares. Choice D is incorrect as the expression remains quadratic in both variables.
See $16x^4 - 81$ as a difference of squares by viewing it as $(4x^2)^2 - 9^2$ , and factor completely over the integers.
Answer: $(2x-3)(2x+3)(4x^2+9)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in $x^4 - y^4$ , if we see it as $(x^2)^2 - (y^2)^2$ , we recognize it's a difference of squares and can factor it as $(x^2 + y^2)(x^2 - y^2)$ , then factor the second part again since it's also a difference of squares! For $16x^4 - 81$ , view it as $(4x^2)^2 - 9^2$ , which factors to $(4x^2 - 9)(4x^2 + 9)$ , and then factor $4x^2 - 9$ further as $(2x - 3)(2x + 3)$ for complete factorization. Choice C correctly applies the difference of squares twice to factor completely to $(2x - 3)(2x + 3)(4x^2 + 9)$ . Excellent pattern recognition! Choice A stops too early without factoring further—remember to check if factors can be broken down more; it's a common step, and you're getting better at it! When you see higher powers like $x^4$ , think: can I view this as a perfect square? $x^4 = (x^2)^2$ , so $x^4 - 1$ becomes $(x^2)^2 - 1^2$ = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For $x^4 - 16 = (x^2 + 4)(x^2 - 4)$ , the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Use structure to factor by grouping: $x^3+4x^2+3x+12$ .
Answer: $(x+4)(x^2+3)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Using structure means looking for patterns you already know—difference of squares, perfect square trinomials, common factors—rather than just trying random factor pairs. It's like having a toolkit of patterns that make factoring much faster when you spot them! For x³ + 4x² + 3x + 12, we can group terms strategically: (x³ + 4x²) + (3x + 12). From the first group, factor out x²: x²(x + 4). From the second group, factor out 3: 3(x + 4). Now we have x²(x + 4) + 3(x + 4). Notice both terms share the common factor (x + 4)! Factoring it out gives us (x + 4)(x² + 3). Choice A correctly shows this factorization as (x + 4)(x² + 3). Excellent pattern recognition! Choice B reverses the factors—while mathematically equivalent due to commutativity, the question asks us to factor by grouping, and the natural result of that process is (x + 4)(x² + 3). When factoring by grouping: (1) Group terms that might share a common factor, (2) Factor out the GCF from each group, (3) Look for a common binomial factor, (4) Factor out that common binomial. Always verify by expanding your answer—it should match the original expression!
Rewrite $9x^2-24x+16$ using its structure as a perfect square trinomial.
Answer: $(3x-4)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! In 9x² - 24x + 16, 9x² = (3x)², 16 = 4², and -24x = -2 · 3x · 4, matching the negative pattern for (3x - 4)². Choice A correctly recognizes the pattern as a perfect square trinomial and rewrites it as (3x - 4)². Excellent pattern recognition! Choice B flips the sign, but verify by checking the middle term—it should be negative here; expanding is a helpful double-check! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Recognize $x^2+14x+49$ as a perfect square trinomial and rewrite it using its structure.
Answer: $(x+7)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² + 14x + 49 is a perfect square trinomial: first term x² is a perfect square (√x² = x), last term 49 is a perfect square (√49 = 7), and the middle term 14x should equal 2 · x · 7 = 14x. Yes! This matches the pattern a² + 2ab + b² with a = x and b = 7. Choice A correctly applies the formula (a + b)² to get (x + 7)². Excellent pattern recognition! Choice B would give us x² - 14x + 49 when expanded, not our original expression—watch those signs carefully! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Rewrite $x^2+14x+49$ using its structure as a perfect square trinomial.
Answer: $(x+7)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² + 14x + 49 fits this pattern. First term: x² is a perfect square (it's x squared). Last term: 49 = 7², so it's a perfect square too. Middle term: 14x should equal 2·x·7 = 14x. Perfect match! Since all signs are positive, this follows the (a + b)² pattern where a = x and b = 7. Choice A correctly identifies this as (x + 7)². Excellent pattern recognition! Choice B would give us (x - 7)² = x² - 14x + 49, which has a negative middle term—not what we started with. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
See $16x^4-81$ as a difference of squares by viewing it as $(4x^2)^2-9^2$ , and factor completely over the integers.
Answer: $(2x-3)(2x+3)(4x^2+9)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in $x^4 - y^4$ , if we see it as $(x^2)^2 - (y^2)^2$ , we recognize it's a difference of squares and can factor it as $(x^2 + y^2)(x^2 - y^2)$ , then factor the second part again since it's also a difference of squares! For $16x^4 - 81$ , view it as $(4x^2)^2 - 9^2$ , which factors to $(4x^2 - 9)(4x^2 + 9)$ , and then factor $4x^2 - 9$ further as $(2x - 3)(2x + 3)$ for complete factorization. Choice C correctly applies the difference of squares twice to factor completely to $(2x - 3)(2x + 3)(4x^2 + 9)$ . Excellent pattern recognition! Choice A stops too early without factoring further—remember to check if factors can be broken down more; it's a common step, and you're getting better at it! When you see higher powers like $x^4$ , think: can I view this as a perfect square? $x^4 = (x^2)^2$ , so $x^4 - 1$ becomes $(x^2)^2 - 1^2$ = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For $x^4 - 16 = (x^2 + 4)(x^2 - 4)$ , the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
To simplify $\frac{x^4 - 16}{x^2 + 4}$ most efficiently, which structural observation is most helpful?
Answer: The numerator can be written as $(x^2)^2 - 4^2$ , revealing a difference of squares The correct answer is B. Recognizing that $x^4 - 16 = (x^2)^2 - 4^2$ shows it's a difference of squares that factors as $(x^2 - 4)(x^2 + 4)$ . Since $x^2 - 4 = (x-2)(x+2)$ , the expression becomes $\frac{(x-2)(x+2)(x^2+4)}{x^2+4} = (x-2)(x+2)$ after cancellation. Choice A suggests unnecessary polynomial division. Choice C incorrectly focuses on finding roots of the denominator. Choice D suggests a less systematic approach than recognizing the difference of squares pattern.
Recognize $x^2-12x+36$ as what type of pattern, and rewrite it in factored form.
Answer: $(x-6)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² - 12x + 36 fits a perfect square pattern. First term: x² is a perfect square. Last term: 36 = 6², also a perfect square. Middle term: -12x should equal -2·x·6 = -12x. Perfect! Since we have positive-negative-positive signs, this matches the (a - b)² pattern where a = x and b = 6. Therefore, x² - 12x + 36 = (x - 6)². Choice A correctly shows this as (x - 6)². Excellent pattern recognition! Choice D would give us (x - 6)(x + 6) = x² - 36, which is missing the middle term entirely—that's a difference of squares, not a perfect square trinomial. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
See $x^4-16$ as $(x^2)^2-4^2$ and factor completely.
Answer: $(x^2+4)(x-2)(x+2)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! For x⁴ - 16, we can view it as (x²)² - 4². Using the difference of squares formula: (x²)² - 4² = (x² + 4)(x² - 4). But we're not done! The factor (x² - 4) is itself a difference of squares since 4 = 2². So x² - 4 = (x + 2)(x - 2). The complete factorization is (x² + 4)(x + 2)(x - 2), which can be written as (x² + 4)(x - 2)(x + 2). Choice C correctly shows this complete factorization. Excellent pattern recognition! Choice A stops after the first application of difference of squares and doesn't recognize that x² - 4 can be factored further. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
See $16x^4-9$ as a difference of squares and factor completely.
Answer: $(4x^2+3)(4x^2-3)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Looking at 16x⁴ - 9, we need to recognize what's being squared: 16x⁴ = (4x²)² and 9 = 3². So we have (4x²)² - 3², which is a difference of squares with a = 4x² and b = 3. Applying the pattern (a² - b² = (a + b)(a - b)) gives us (4x² + 3)(4x² - 3). Choice A correctly recognizes 16x⁴ - 9 as (4x²)² - 3² and applies the difference of squares formula to get (4x² + 3)(4x² - 3). Excellent pattern recognition! Choice D would only work if we had 16x² - 9, not 16x⁴—watch those exponents carefully! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first.
See $x^4-81$ as a difference of squares by viewing it as $(x^2)^2-9^2$ , and factor completely.
Answer: $\left(x^2+9\right)\left(x-3\right)\left(x+3\right)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). Here we have x⁴ - 81, which we can view as (x²)² - 9². Applying the difference of squares formula: (x²)² - 9² = (x² + 9)(x² - 9). But wait—we're not done! The factor (x² - 9) is itself a difference of squares: x² - 9 = (x + 3)(x - 3). So the complete factorization is (x² + 9)(x + 3)(x - 3). Choice C correctly shows this complete factorization with all three factors. Excellent pattern recognition! Choice A stops too early—it factors the first difference of squares but doesn't recognize that x² - 9 can be factored further. When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
What is $x^4-16$ when factored completely? (Hint: view it as $(x^2)^2-4^2$ .)
Answer: $(x-2)(x+2)(x^2+4)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Here, x⁴ - 16 is (x²)² - 4², factoring to (x² - 4)(x² + 4), and then x² - 4 is another difference of squares: (x - 2)(x + 2). Choice B correctly factors completely to (x - 2)(x + 2)(x² + 4). Excellent pattern recognition! Choice D might come from thinking it's a perfect square, but expand it to check—it gives extra terms; keep practicing by verifying your factors multiply back correctly! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Use structure to rewrite $x^2+10x+25$ as a single squared binomial.
Answer: $(x+5)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² + 10x + 25 is a perfect square trinomial. First term: x² is clearly a perfect square. Last term: 25 = 5², also a perfect square. Middle term: 10x should equal 2·x·5 = 10x. Yes! All three conditions are met, and all signs are positive, so this follows the (a + b)² pattern where a = x and b = 5. Therefore, x² + 10x + 25 = (x + 5)². Choice C correctly identifies this as (x + 5)². Excellent pattern recognition! Choice A would give us (x + 5)(x - 5) = x² - 25, which is a difference of squares with no middle term—not what we started with. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
The expression $(x + 2)^3 - (x + 2)$ can be factored most efficiently by recognizing which structure?
Answer: Factor out $(x + 2)$ to get $(x + 2)[(x + 2)^2 - 1]$ When you encounter expressions with repeated factors like $(x + 2)^3 - (x + 2)$ , look for opportunities to factor out common terms before doing any complex algebra. This approach saves time and reduces the chance of calculation errors. The key insight here is recognizing that both terms share the factor $(x + 2)$ . By factoring this out, you get $(x + 2)[(x + 2)^2 - 1]$ . Now you have a much simpler expression to work with, and you can further factor $(x + 2)^2 - 1$ as a difference of squares: $(x + 2 - 1)(x + 2 + 1) = (x + 1)(x + 3)$ . So the complete factorization becomes $(x + 2)(x + 1)(x + 3)$ . Option A would work but is unnecessarily complicated—expanding $(x + 2)^3$ creates a mess of terms that you'd then have to factor again. Option B misapplies the difference of cubes formula, which requires the form $a^3 - b^3$ , but here we have $a^3 - a$ , not $a^3 - b^3$ . Option C suggests synthetic division, which is overkill for this factoring problem and assumes you already know $x = -2$ is a root. Remember this pattern: when you see expressions with the same base raised to different powers, always check if you can factor out the lowest power first. This "factor-first" strategy often reveals the structure more clearly than expanding everything out.
Factor $x^2-10x+25$ by recognizing it as a perfect square trinomial.
Answer: $(x-5)^2$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! In x² - 10x + 25, we have x² = (x)², 25 = (5)², and -10x = -2 · x · 5, matching the negative version, so it's (x - 5)². Choice A correctly recognizes the pattern as a perfect square trinomial and applies the formula (a - b)². Excellent pattern recognition! If you chose D, that's difference of squares, but notice the middle term means it's not just two terms being subtracted—look for that trinomial structure. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
See $x^4-y^4$ as $(x^2)^2-(y^2)^2$ and factor completely.
Answer: $(x-y)(x+y)(x^2+y^2)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! So for x⁴ - y⁴, first (x² + y²)(x² - y²), then factor x² - y² to (x + y)(x - y), giving (x + y)(x - y)(x² + y²) completely. Choice B correctly applies the formula and factors completely to get (x - y)(x + y)(x² + y²). Excellent pattern recognition! If you stopped at A, good start, but check for further factoring in each binomial. When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Which form best reveals why $49x^2 - 14x + 1$ and $49x^2 + 14x + 1$ have a special relationship?
Answer: $(7x)^2 \pm 2(7x)(1) + 1^2$ shows both are perfect square trinomials The correct answer is B. Writing both expressions as $(7x)^2 \pm 2(7x)(1) + 1^2$ reveals they are both perfect square trinomials: $(7x - 1)^2$ and $(7x + 1)^2$ respectively. This structural insight shows their special relationship as conjugate squares and explains why their product would eliminate the middle term: $(7x-1)^2(7x+1)^2 = [(7x)^2 - 1]^2$ . Choice A merely states the obvious difference. Choice C doesn't reveal the perfect square structure. Choice D creates an unhelpful form that obscures the structure.
See $16x^2 - 9$ as a difference of squares and rewrite it as a product of two binomials.
Answer: $(4x+3)(4x-3)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern $a^2 - b^2 = (a + b)(a - b)$ is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, $x^2 - 9 = (x)^2 - (3)^2 = (x + 3)(x - 3)$ . Let's examine $16x^2 - 9$ for the difference of squares pattern. First term: $16x^2 = (4x)^2$ , which is a perfect square. Second term: $9 = 3^2$ , also a perfect square. They're being subtracted with no middle term—we have a difference of squares! Applying the formula: $(4x)^2 - 3^2 = (4x + 3)(4x - 3)$ . Choice B correctly identifies this as $(4x + 3)(4x - 3)$ . Excellent pattern recognition! Choice A shows $(16x - 3)(x + 3)$ , but if you multiply this out, you get $16x^2 + 48x - 3x - 9 = 16x^2 + 45x - 9$ , which has a middle term our original doesn't have. To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like $x^2$ , 9, $4x^2$ , 25)? If yes to all three, you've got $a^2 - b^2$ , which factors as $(a + b)(a - b)$ . Try it with $x^2 - 16$ : yes two terms, yes subtraction, yes both perfect squares → $(x + 4)(x - 4)$ . Boom!
Rewrite $x^2-9$ using its structure as a difference of squares.
Answer: $(x+3)(x-3)$ This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). For x² - 9, it's simply (x)² - (3)², so factor as (x + 3)(x - 3). Choice C correctly recognizes the pattern as difference of squares and applies the formula (a + b)(a - b). Excellent pattern recognition! If you picked A or D, you might have seen it as a perfect square, but there's no middle term here—perfect squares are trinomials with specific signs. To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like x², 9, 4x², 25)? If yes to all three, you've got a² - b², which factors as (a + b)(a - b). Try it with x² - 16: yes two terms, yes subtraction, yes both perfect squares → (x + 4)(x - 4). Boom!

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