This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive $a$ = both ends up, (2) even degree + negative $a$ = both ends down, (3) odd degree + positive $a$ = left down, right up, (4) odd degree + negative $a$ = left up, right down. Memorize these four! For $g(x) = -(x+3)(x-2)^2(x-4)$, zeros are $x = -3$ (mult 1, crosses), $x = 2$ (mult 2, touches), $x = 4$ (mult 1, crosses); degree 4 even with negative leading $-x^4$, so both to $-\infty$. Choice A correctly identifies the zeros with proper behaviors and the both-down end behavior. In choice C, the end behavior is listed as both up, but that's for positive leading coefficient—remember to check the sign! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate $f(0)$, (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior. You don't need exact turning points—just show the zeros, their behavior, and where the graph ends up as $x \to \pm\infty$!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $$p(x) = 2x^4 - 100x^3 + 500x - 1000$$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive $a$ = both ends up, (2) even degree + negative $a$ = both ends down, (3) odd degree + positive $a$ = left down, right up, (4) odd degree + negative $a$ = left up, right down. Memorize these four! For $p(x) = 3x^5 - 2x^3 + 7x - 1$, the leading term is $3x^5$ (degree 5 odd, positive), so as $x \to -\infty$, $p(x) \to -\infty$ and as $x \to \infty$, $p(x) \to \infty$. Choice B correctly describes this left-down, right-up behavior for odd positive leading. A choice like A reverses it, but remember the pattern for positive odd: left down, right up—you're getting the hang of it! End behavior shortcut: (1) find degree $n$—count highest power, (2) find sign of leading coefficient $a$—look at coefficient of $x^n$ term, (3) apply pattern: even $n$ = both ends match (up if $a > 0$, down if $a < 0$); odd $n$ = ends opposite (if $a > 0$: ↓↑, if $a < 0$: ↑↓). Example: $-3x^5 + 100x^2 - 50$ has degree 5 (odd), $a = -3$ (negative), so left up, right down. Ignore all other terms—only the leading term matters for end behavior!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny. The four end behavior patterns are: (1) even degree + positive a = both ends up, (2) even degree + negative a = both ends down, (3) odd degree + positive a = left down, right up, (4) odd degree + negative a = left up, right down. Memorize these four! For p(x) = $-x^3$ + $6x^2$ - 9x = $-x(x-3)^2$, zeros x = 0 (mult 1), x = 3 (mult 2); degree 3 odd negative, so left +∞, right -∞. Choice A correctly factors, gives multiplicities, and matches the end behavior. Choice C swaps multiplicities and reverses end behavior, but odd negative is left up right down—practice the patterns! End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). Example: -3x⁵ + 100x² - 50 has degree 5 (odd), a = -3 (negative), so left up, right down. Ignore all other terms—only the leading term matters for end behavior!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. Graphing a polynomial requires two main elements: (1) zeros (found from factored form by setting each factor equal to zero) with their multiplicities determining whether the graph crosses (odd multiplicity) or touches (even multiplicity) at each zero, and (2) end behavior (determined by the leading term's degree and sign). For example, p(x) = -2x³ has degree 3 (odd) and leading coefficient -2 (negative), so as x → -∞, p(x) → +∞ (left end up), and as x → +∞, p(x) → -∞ (right end down). These two elements give you the skeleton of the graph! For f(x) = $x^4$ - $5x^2$ + 4 = $(x^2$$-1)(x^2$-4) = (x-1)(x+1)(x-2)(x+2), zeros ±1, ±2 (all mult 1, crosses); degree 4 even positive, both to +∞. Choice A correctly factors and shows all zeros crossing with both-up end. Choice D has the right factoring but wrong end (both down), but positive leading means up—keep that sign in mind! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior. You don't need exact turning points—just show the zeros, their behavior, and where the graph ends up as x → ±∞!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive $a$ = both ends up, (2) even degree + negative $a$ = both ends down, (3) odd degree + positive $a$ = left down, right up, (4) odd degree + negative $a$ = left up, right down—memorize these four! For $q(x) = -x^5 + 4x^3 - x$, the leading term is $-x^5$ (degree 5 odd, negative coefficient), so as $x \to -\infty$, $q(x) \to +\infty$, and as $x \to +\infty$, $q(x) \to -\infty$ (left up, right down). Choice B correctly describes the end behavior for this odd negative degree polynomial. A distractor like choice C might treat it as even degree, but confirm the highest power is odd here. End behavior shortcut: (1) find degree $n$—count highest power, (2) find sign of leading coefficient $a$—look at coefficient of $x^n$ term, (3) apply pattern: even $n$ = both ends match (up if $a > 0$, down if $a < 0$); odd $n$ = ends opposite (if $a > 0$: ↓↑, if $a < 0$: ↑↓)—for example, $-3x^5 + 100x^2 - 50$ has degree 5 (odd), $a = -3$ (negative), so left up, right down—ignore all other terms! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate $f(0)$, (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior—keep it up!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive a = both ends up, (2) even degree + negative a = both ends down, (3) odd degree + positive a = left down, right up, (4) odd degree + negative a = left up, right down—memorize these four! For g(x) = $5x^4$ - $3x^2$ + 7, the leading term is $5x^4$ (degree 4 even, positive coefficient), so as x → ±∞, g(x) → +∞ (both ends up). Choice C correctly identifies the end behavior as both ends approaching positive infinity, aligning with the even positive pattern. A distractor like choice A might confuse it with odd degree behavior, but always ignore lower terms and focus on the highest power. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓)—ignore all other terms—only the leading term matters for end behavior! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior—you're building strong skills!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. Graphing a polynomial requires two main elements: (1) zeros (found from factored form by setting each factor equal to zero) with their multiplicities determining whether the graph crosses (odd multiplicity) or touches (even multiplicity) at each zero, and (2) end behavior (determined by the leading term's degree and sign)—for example, p(x) = -2x³ has degree 3 (odd) and leading coefficient -2 (negative), so as x → -∞, p(x) → +∞ (left end up), and as x → +∞, p(x) → -∞ (right end down). For f(x) = (x-2)(x+1)(x+4), the zeros are x = 2, -1, -4 (all multiplicity 1, odd, cross); degree 3 (odd) with positive leading coefficient $x^3$, so left down, right up. Choice A correctly lists the zeros (noting x = -4, -1, 2) with crossing and the end behavior for odd positive degree. A distractor like choice B might reverse the end behavior, but recall that positive odd degrees go down on left and up on right. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior— you're mastering this!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large $|x|$, this term dominates all others: in $p(x) = 2x^4 - 100x^3 + 500x - 1000$, for $x = 1000$, the $2x^4$ term equals 2 trillion while other terms are relatively tiny—the four end behavior patterns are: (1) even degree + positive a = both ends up, (2) even degree + negative a = both ends down, (3) odd degree + positive a = left down, right up, (4) odd degree + negative a = left up, right down—memorize these four! For $p(x) = x(x-2)^2(x+1)$, the zeros are $x = 0$ (multiplicity 1, odd, crosses), $x = 2$ (multiplicity 2, even, touches), $x = -1$ (multiplicity 1, odd, crosses); degree 4 (even) with positive leading coefficient $x^4$, so both ends up. Choice A correctly identifies the zeros with their crossing/touching behaviors and the end behavior matching the even positive pattern. A distractor like choice C might incorrectly apply odd degree end behavior, but always sum the multiplicities for the total degree. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓)—ignore all other terms—only the leading term matters for end behavior! The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior—excellent work!

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. End behavior depends ONLY on the leading term $ax^n$, because for large |x|, this term dominates all others: in p(x) = 2x⁴ - 100x³ + 500x - 1000, for x = 1000, the 2x⁴ term equals 2 trillion while other terms are relatively tiny. For g(x) = $x(x-2)^2$(x+3), zeros are x = 0 (cross), x = 2 (touch), x = -3 (cross), y-intercept is (0,0), and degree 4 even positive means left up, right up. Choice A correctly shows the zeros with proper cross/touch, y-intercept, and both ends up. Choice D fails by incorrectly using odd positive end behavior (left down, right up) instead of even positive. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior.

This question tests your ability to graph polynomial functions by identifying zeros from factorizations and determining end behavior from the leading term's degree and coefficient. Graphing a polynomial requires two main elements: (1) zeros (found from factored form by setting each factor equal to zero) with their multiplicities determining whether the graph crosses (odd multiplicity) or touches (even multiplicity) at each zero, and (2) end behavior (determined by the leading term's degree and sign). For h(x) = $-3x^5$ + $2x^3$ - 7x + 1, the leading term is $-3x^5$ (odd degree, negative coefficient), so as x → -∞, h(x) → +∞ and as x → +∞, h(x) → -∞. Choice B correctly describes this end behavior for odd degree negative. A distractor like Choice A reverses it to odd positive, which would require a positive leading coefficient. End behavior shortcut: (1) find degree n—count highest power, (2) find sign of leading coefficient a—look at coefficient of $x^n$ term, (3) apply pattern: even n = both ends match (up if a > 0, down if a < 0); odd n = ends opposite (if a > 0: ↓↑, if a < 0: ↑↓). The complete polynomial graphing checklist: (1) Find zeros: set each factor equal to zero (watch signs!), (2) Determine multiplicity: count factor appearances, note cross (odd) or touch (even) at each zero, (3) Find y-intercept: evaluate f(0), (4) Determine end behavior: degree + leading coefficient sign, (5) Plot zeros and y-intercept on axes, (6) Sketch smooth curve through/touching zeros with correct end behavior.