Polynomial Functions - ACT Math

1. What is the leading coefficient of $7x^3 + 5x^2 - 4x + 6$?
   Answer: 7 The leading coefficient is the coefficient of the term with the highest degree. In the polynomial 7x³ + 5x² - 4x + 6, the highest degree term is 7x³ with degree 3. The coefficient of this term is 7, which is the leading coefficient. Choice B might incorrectly identify the coefficient of the x² term instead.
2. An expression for revenue is $x(x-5) + 2(x+1)$. Which polynomial is equivalent to this expression in standard form?
   Answer: $x^2 - 3x + 2$ To simplify x(x-5) + 2(x+1), first distribute: x(x-5) = x² - 5x and 2(x+1) = 2x + 2. Then add: x² - 5x + 2x + 2 = x² + (-5x + 2x) + 2. Combine like terms: x² - 3x + 2. Choice B has the wrong middle term coefficient.
3. A polynomial expression used for a pattern is $(x+1)^2 - 4x$. Which polynomial is equivalent to this expression in standard form?
   Answer: $x^2 - 2x + 1$ To simplify (x+1)² - 4x, first expand the square: (x+1)² = x² + 2x + 1. Then subtract 4x: x² + 2x + 1 - 4x = x² + (2x - 4x) + 1. Combine like terms: x² - 2x + 1. Choice B has the wrong sign on the middle term.
4. What is the degree of the polynomial $x^2 + 2x + 1$?
   Answer: 2 The degree of a polynomial is the highest power of the variable. In the polynomial x² + 2x + 1, we identify the highest exponent among all terms. The terms have powers 2, 1, and 0 respectively, so the highest power is 2. Choice C might confuse the number of terms with the degree.
5. What is $f(1)$ for the polynomial $f(x) = 4x^2 - 2x + 1$?
   Answer: 3 We need to evaluate f(1) by substituting x = 1 into f(x) = 4x² - 2x + 1. Substituting carefully: f(1) = 4(1)² - 2(1) + 1 = 4(1) - 2 + 1 = 4 - 2 + 1 = 3.
6. Which polynomial is equivalent to $3(x^2 - 2x + 1)$?
   Answer: $3x^2 - 6x + 3$ This tests the distributive property: a factor outside parentheses multiplies every term inside, and it keeps each term's sign. Multiplying term by term gives $3 \cdot x^2 = 3x^2$, $3 \cdot (-2x) = -6x$, and $3 \cdot 1 = 3$, so the equivalent polynomial is $3x^2 - 6x + 3$. The version $3x^2 + 6x + 3$ comes from dropping the minus sign on the middle term, $3x^2 - 2x + 1$ comes from multiplying only the leading term and leaving the rest untouched, and $3x^2 - 2x + 3$ comes from distributing to the first and last terms but forgetting the middle one. When a single factor sits outside parentheses, count the terms inside first and make sure your answer shows that same number of products, each with its original sign.
7. A polynomial is given by $t(x) = 7x - 3 + 2x^3$. What is the degree of $t(x)$?
   Answer: $3$ The degree of a polynomial is the highest exponent of the variable. In t(x) = 7x - 3 + 2x³, rewrite in standard form as t(x) = 2x³ + 7x - 3. The terms have exponents: 2x³ has exponent 3, 7x has exponent 1, and -3 has exponent 0. The highest exponent is 3, so the degree is 3.
8. For a fundraising event, the profit (in dollars) after selling $x$ tickets is modeled by the polynomial function $P(x)=2x^3-3x^2+4x-5$. What is $P(-2)$?
   Answer: $-41$ We need to evaluate P(x) = 2x³ - 3x² + 4x - 5 at x = -2. Substituting x = -2: P(-2) = 2(-2)³ - 3(-2)² + 4(-2) - 5 = 2(-8) - 3(4) + 4(-2) - 5 = -16 - 12 - 8 - 5. Calculating step by step: -16 - 12 = -28, then -28 - 8 = -36, and finally -36 - 5 = -41. Choice A incorrectly evaluates as -25, likely from a sign error in the cubic term.
9. A polynomial is written as $w(x) = 3x^2 + 0x + 4$. What is the degree of $w(x)$?
   Answer: $2$ The degree of a polynomial is the highest exponent of the variable. In w(x) = 3x² + 0x + 4, the terms have exponents: 3x² has exponent 2, 0x has exponent 1, and 4 has exponent 0. The highest exponent is 2, so the degree is 2. Choice A might confuse the constant term with the degree.
10. What is the leading coefficient of $-x^3 + 4x^2 - x$?
    Answer: -1 The leading coefficient is the coefficient of the term with the highest degree. In the polynomial -x³ + 4x² - x, the highest degree term is -x³ with degree 3. Since -x³ = (-1)x³, the coefficient of this term is -1, which is the leading coefficient. Choice C might incorrectly identify the coefficient of the x² term.
11. A company models weekly profit (in thousands of dollars) by the polynomial $P(x)=-x^3+4x-1$, where $x$ is the number of weeks since launch. What is the degree of $P(x)$?
    Answer: $3$ The degree of a polynomial is the highest power of x. In P(x) = -x³ + 4x - 1, the highest power is x³, so the degree is 3. The coefficient -1 in front of x³ is the leading coefficient, not the degree.
12. Which polynomial is equivalent to $x^2 - 4x + 4$?
    Answer: $(x-2)^2$ We need to factor x² - 4x + 4. This is a perfect square trinomial of the form a² - 2ab + b² = (a - b)². Here, we have x² - 2(x)(2) + 2² = x² - 4x + 4 = (x - 2)². We can verify by expanding: (x - 2)² = x² - 4x + 4.
13. Which polynomial is equivalent to $x(x - 1) + 2(x + 1)$?
    Answer: $x^2 + x + 2$ We need to simplify x(x - 1) + 2(x + 1) by distributing and combining like terms. First distribute: x(x - 1) = x² - x and 2(x + 1) = 2x + 2. Combine: x² - x + 2x + 2 = x² + (-x + 2x) + 2 = x² + x + 2. Choice B would result from sign errors when combining the x terms.
14. A polynomial is given by $a(x) = 9 - 5x + x^3$. What is the degree of $a(x)$?
    Answer: $3$ The degree of a polynomial is the highest exponent of the variable. In a(x) = 9 - 5x + x³, rewrite in standard form as a(x) = x³ - 5x + 9. The terms have exponents: x³ has exponent 3, -5x has exponent 1, and 9 has exponent 0. The highest exponent is 3, so the degree is 3.
15. What is the leading coefficient of the polynomial $4x - 7$?
    Answer: 4 We need to identify the leading coefficient of 4x - 7. The leading coefficient is the coefficient of the term with the highest degree. The term 4x has degree 1 (the highest degree), and the term -7 has degree 0. Therefore, the leading coefficient is 4.
16. A polynomial function is $y(x) = 6x^3 + x^2 - 2$. What is the leading coefficient of $y(x)$?
    Answer: $6$ The leading coefficient is the coefficient of the term with the highest degree. In y(x) = 6x³ + x² - 2, identify the highest degree term: 6x³ has degree 3, which is the highest among all terms. The coefficient of this term is 6, making it the leading coefficient. Choice A might confuse the degree with the coefficient.
17. A company uses the polynomial $C(x)= -4x^2+3x+9$ to model cost. What is the leading coefficient of $C(x)$?
    Answer: $-4$ The leading coefficient is the coefficient of the term with the highest degree. In C(x) = -4x² + 3x + 9, the highest degree term is -4x² with degree 2. The coefficient of this term is -4, so the leading coefficient is -4. Choice A incorrectly identifies 9 as the leading coefficient, confusing it with the constant term.
18. Which polynomial is equivalent to $(2x - 3)^2$?
    Answer: $4x^2 - 12x + 9$ We need to expand (2x - 3)² using the perfect square formula. Using (a - b)² = a² - 2ab + b², where a = 2x and b = 3: (2x - 3)² = (2x)² - 2(2x)(3) + 3² = 4x² - 12x + 9.
19. Evaluate $f(3)$ for $f(x) = x^2 - 4x + 3$.
    Answer: 0 We need to evaluate f(3) for f(x) = x² - 4x + 3. Substitute x = 3: f(3) = (3)² - 4(3) + 3. Calculate each term: 9 - 12 + 3 = 0. This shows that x = 3 is a root of this polynomial since f(3) = 0. Choice B might result from forgetting to include the constant term.
20. A polynomial $f(x)=x^3-x$ models a changing quantity. Which statement best describes the end behavior of $f(x)$?
    Answer: As $x\to\infty$, $f(x)\to \infty$; as $x\to -\infty$, $f(x)\to -\infty$. For f(x) = x³ - x, the leading term x³ has positive coefficient 1 and odd degree 3. For odd-degree polynomials with positive leading coefficient: as x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞. This matches the standard end behavior pattern.

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