ACT Math Flashcards: Polynomial Functions

What this deck covers

This deck focuses on Polynomial Functions, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

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: What happens at $x=r$ when a zero has odd multiplicity?

Answer: The graph crosses the $x$-axis at $x=r$. Odd multiplicity means the graph passes through the $x$-axis.

Flashcard 2: What is the conjugate root rule for polynomials with real coefficients?

Answer: If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials always come in conjugate pairs.

Flashcard 3: Identify the leading coefficient of $f(x)=-4x^6+2x-1$.

Answer: Leading coefficient $=-4$. The coefficient of the highest degree term is the leading coefficient.

Flashcard 4: What is the degree of a polynomial written as $a_nx^n+\cdots+a_0$ with $a_n\ne 0$?

Answer: Degree $=n$. The degree equals the highest power of $x$ when $a_n \neq 0$.

Flashcard 5: What is the sum of the roots of $x^2 - 4x + 3$?

Answer: The sum is 4. For $x^2 - 4x + 3$, sum equals $-b/a = 4$.

Flashcard 6: Identify the constant term in $3x^4 - 2x + 7$.

Answer: The constant term is 7. The term without any variable.

Flashcard 7: What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n<0$?

Answer: As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like behavior.

Flashcard 8: What is the vertex $x$-coordinate of $f(x)=ax^2+bx+c$?

Answer: $x_v=-\frac{b}{2a}$. The vertex $x$-coordinate is the axis of symmetry.

Flashcard 9: What is the discriminant of $ax^2+bx+c$ used to classify real roots?

Answer: $\Delta=b^2-4ac$. The discriminant determines the nature of quadratic roots.

Flashcard 10: Factor $x^2 + 5x + 6$ completely.

Answer: $(x + 2)(x + 3)$. Find factors of 6 that sum to 5.

Flashcard 11: What is the constant term in $x^3 + 2x + 5$?

Answer: The constant term is 5. The term without any variable.

Flashcard 12: What does $\Delta=b^2-4ac>0$ imply about the roots of $ax^2+bx+c$?

Answer: Two distinct real roots. Positive discriminant means the parabola crosses the $x$-axis twice.

Flashcard 13: Identify the constant term in $3x^4 - 2x + 7$.

Answer: The constant term is 7. The term without any variable.

Flashcard 14: What happens at $x=r$ when a zero has odd multiplicity?

Answer: The graph crosses the $x$-axis at $x=r$. Odd multiplicity means the graph passes through the $x$-axis.

Flashcard 15: Write the polynomial $2x^3 + 3x - x^3$ in standard form.

Answer: $x^3 + 3x$. Combine like terms and order by degree.

Flashcard 16: Find the product of $2x$ and $(3x + 4)$.

Answer: $6x^2 + 8x$. Distribute $2x$ to each term.

Flashcard 17: What is the greatest common factor of $4x^3$ and $6x^2$?

Answer: $2x^2$. Factor out $2x^2$ from both terms.

Flashcard 18: What is the degree of a constant polynomial?

Answer: The degree is 0. A constant has no variables.

Flashcard 19: What does $\Delta=b^2-4ac>0$ imply about the roots of $ax^2+bx+c$?

Answer: Two distinct real roots. Positive discriminant means the parabola crosses the $x$-axis twice.

Flashcard 20: What is the maximum possible number of turning points of a degree $n$ polynomial?

Answer: At most $n-1$ turning points. Each turning point reduces the maximum by one from the degree.

Flashcard 21: Find the maximum number of turning points for a degree $6$ polynomial.

Answer: At most $5$ turning points. A degree $6$ polynomial has at most $6-1=5$ turning points.

Flashcard 22: What is the degree of a constant polynomial?

Answer: The degree is 0. A constant has no variables.

Flashcard 23: What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n<0$?

Answer: As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like behavior.

Flashcard 24: Identify the leading coefficient of $f(x)=-4x^6+2x-1$.

Answer: Leading coefficient $=-4$. The coefficient of the highest degree term is the leading coefficient.

Flashcard 25: What is the term for a polynomial with one term?

Answer: Monomial. A polynomial with exactly one term.

Flashcard 26: Find the roots of the polynomial $x^2 - x - 6$.

Answer: $x = 3, x = -2$. Factor as $(x-3)(x+2) = 0$.

Flashcard 27: What is the leading coefficient of $-5x^4 + 3x^3$?

Answer: The leading coefficient is $-5$. The coefficient of the highest degree term.

Flashcard 28: Factor $x^2 - 4x + 4$ completely.

Answer: $(x - 2)^2$. Perfect square trinomial pattern.

Flashcard 29: How do you factor $x^2 - 9$?

Answer: $(x + 3)(x - 3)$. Difference of squares: $x^2 - 3^2$.

Flashcard 30: How do you factor $x^2 - 9$?

Answer: $(x + 3)(x - 3)$. Difference of squares: $x^2 - 3^2$.

Flashcard 31: Identify the leading coefficient of $7x^5 - 3x^3 + x$.

Answer: The leading coefficient is 7. The coefficient of the highest degree term.

Flashcard 32: Find the result of multiplying $x^2$ by $3x$.

Answer: $3x^3$. Multiply coefficients and add exponents.

Flashcard 33: Identify the multiplicity of the zero $x=2$ for $f(x)=(x-2)^3(x+1)$.

Answer: Multiplicity $=3$. The exponent of $(x-2)$ gives the multiplicity.

Flashcard 34: Write the polynomial $2x^3 + 3x - x^3$ in standard form.

Answer: $x^3 + 3x$. Combine like terms and order by degree.

Flashcard 35: What is the degree of the polynomial $5x^2y^3 + 3xy - 4$?

Answer: The degree is 5. The sum of exponents in $x^2y^3$ is $2+3=5$.

Flashcard 36: What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?

Answer: One real double root. Zero discriminant means the parabola touches the $x$-axis once.

Flashcard 37: What is the degree of the polynomial $x^2y^3 + xy^2$?

Answer: The degree is 5. Highest degree term has exponents $2+3=5$.

Flashcard 38: Identify the possible rational zeros of $f(x)=2x^3-3x^2-8x+12$.

Answer: $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12,\pm\frac{1}{2},\pm\frac{3}{2}$. Use $\pm\frac{p}{q}$ where $p$ divides $12$ and $q$ divides $2$.

Flashcard 39: What is the leading coefficient of $f(x)=a_nx^n+\cdots+a_0$?

Answer: Leading coefficient $=a_n$. The coefficient of the highest degree term determines the leading coefficient.

Flashcard 40: Identify whether the graph crosses or touches at $x=1$ for $f(x)=(x-1)^4(x+2)$.

Answer: Touches and turns at $x=1$. Even multiplicity $(x-1)^4$ means the graph touches and turns.

Flashcard 41: What is the vertex $x$-coordinate of $f(x)=ax^2+bx+c$?

Answer: $x_v=-\frac{b}{2a}$. The vertex $x$-coordinate is the axis of symmetry.

Flashcard 42: What is the Factor Theorem stated in terms of $f(r)$ and $(x-r)$?

Answer: $f(r)=0\iff (x-r)$ is a factor of $f(x)$. The Factor Theorem connects zeros to linear factors.

Flashcard 43: Identify the form: $ax^2 + bx + c$.

Answer: Quadratic form. Standard form of a degree-2 polynomial.

Flashcard 44: Identify the zeroes of $x^2 - 5x + 6$.

Answer: $x = 2, x = 3$. Set $(x-2)(x-3) = 0$ and solve.

Flashcard 45: Find the remainder when $f(x)=x^3-4x+1$ is divided by $(x-2)$.

Answer: $f(2)=1$. By the Remainder Theorem, $f(2) = 8 - 8 + 1 = 1$.

Flashcard 46: What is the axis of symmetry of $f(x)=ax^2+bx+c$?

Answer: $x=-\frac{b}{2a}$. The axis of symmetry passes through the vertex of the parabola.

Flashcard 47: What is the sum of the zeros of $ax^2+bx+c$ (counting multiplicity)?

Answer: $r_1+r_2=-\frac{b}{a}$. Vieta's formula relates coefficients to sums of roots.

Flashcard 48: What is the formula for the sum of cubes?

Answer: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Factorization formula for $a^3 + b^3$.

Flashcard 49: What is the leading coefficient of $f(x)=a_nx^n+\cdots+a_0$?

Answer: Leading coefficient $=a_n$. The coefficient of the highest degree term determines the leading coefficient.

Flashcard 50: What does it mean for $r$ to be a zero (root) of $f(x)$?

Answer: $f(r)=0$. A zero is an $x$-value where the function equals zero.

Flashcard 51: Identify the multiplicity of the zero $x=2$ for $f(x)=(x-2)^3(x+1)$.

Answer: Multiplicity $=3$. The exponent of $(x-2)$ gives the multiplicity.

Flashcard 52: What is the discriminant of $ax^2+bx+c$ used to classify real roots?

Answer: $\Delta=b^2-4ac$. The discriminant determines the nature of quadratic roots.

Flashcard 53: What is the Rational Root Theorem for possible rational zeros of $f(x)$?

Answer: $\frac{p}{q}$ with $p \mid a_0$ and $q \mid a_n$. Rational zeros must have numerator dividing constant term, denominator dividing leading coefficient.

Flashcard 54: What is the degree of the polynomial $x^2y^3 + xy^2$?

Answer: The degree is 5. Highest degree term has exponents $2+3=5$.

Flashcard 55: What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n>0$?

Answer: As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient goes from bottom-left to top-right.

Flashcard 56: What is the standard form of a polynomial?

Answer: Terms are ordered by descending powers of $x$. Powers decrease from left to right.

Flashcard 57: Find the discriminant of $f(x)=x^2-6x+13$ and classify the real roots.

Answer: $\Delta=-16$, no real roots. Calculate $\Delta = 36 - 52 = -16 < 0$, so no real roots.

Flashcard 58: State the term for a polynomial with two terms.

Answer: Binomial. A polynomial with exactly two terms.

Flashcard 59: State the formula for the difference of squares.

Answer: $a^2 - b^2 = (a + b)(a - b)$. Factor pattern for $a^2 - b^2$.

Flashcard 60: What is the multiplicity rule for a factor $(x-r)^k$ of $f(x)$?

Answer: Zero $r$ has multiplicity $k$. The exponent of the factor equals the multiplicity of the zero.

Flashcard 61: How do you write $x^3 - 27$ as a product?

Answer: $(x - 3)(x^2 + 3x + 9)$. Difference of cubes: $x^3 - 3^3$.

Flashcard 62: Identify the degree of $f(x)=7x^5-3x^2+9$.

Answer: Degree $=5$. The highest power term determines the degree.

Flashcard 63: Subtract $4x + 7$ from $6x + 3$.

Answer: $2x - 4$. $(6x + 3) - (4x + 7) = 2x - 4$.

Flashcard 64: What is the value of $f(0)$ for $f(x)=a_nx^n+\cdots+a_0$?

Answer: $f(0)=a_0$. Substituting $x = 0$ eliminates all terms except the constant.

Flashcard 65: Subtract $4x + 7$ from $6x + 3$.

Answer: $2x - 4$. $(6x + 3) - (4x + 7) = 2x - 4$.

Flashcard 66: What is the standard form of a polynomial function in $x$ of degree $n$?

Answer: $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, $a_n\ne 0$. Standard polynomial form with highest degree term first and nonzero leading coefficient.

Flashcard 67: Find the remainder when $f(x)=x^3-4x+1$ is divided by $(x-2)$.

Answer: $f(2)=1$. By the Remainder Theorem, $f(2) = 8 - 8 + 1 = 1$.

Flashcard 68: Identify the leading coefficient of $7x^5 - 3x^3 + x$.

Answer: The leading coefficient is 7. The coefficient of the highest degree term.

Flashcard 69: Factor $x^2 - 4x + 4$ completely.

Answer: $(x - 2)^2$. Perfect square trinomial pattern.

Flashcard 70: What is the maximum possible number of turning points of a degree $n$ polynomial?

Answer: At most $n-1$ turning points. Each turning point reduces the maximum by one from the degree.

Flashcard 71: State the formula for the difference of cubes.

Answer: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Factorization formula for $a^3 - b^3$.

Flashcard 72: What is the degree of a polynomial written as $a_nx^n+\cdots+a_0$ with $a_n\ne 0$?

Answer: Degree $=n$. The degree equals the highest power of $x$ when $a_n \neq 0$.

Flashcard 73: What is the maximum possible number of real zeros of a degree $n$ polynomial?

Answer: At most $n$ real zeros. The Fundamental Theorem of Algebra limits real zeros to at most the degree.

Flashcard 74: What is the conjugate root rule for polynomials with real coefficients?

Answer: If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials always come in conjugate pairs.

Flashcard 75: What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n<0$?

Answer: As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Odd degree with negative leading coefficient goes from top-left to bottom-right.

Flashcard 76: Find the axis of symmetry of $f(x)=2x^2-8x+1$.

Answer: $x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2$.

Flashcard 77: Find the sum of $2x^3 + x^2$ and $3x^3 - 5x^2$.

Answer: $5x^3 - 4x^2$. Combine like terms: $(2+3)x^3 + (1-5)x^2$.

Flashcard 78: Identify the type of polynomial: $x^3 - 2x^2 + x - 5$.

Answer: Cubic polynomial. The highest degree is 3.

Flashcard 79: Identify whether $(x-3)$ is a factor of $f(x)=x^2-6x+9$.

Answer: Yes, because $f(3)=0$. Since $f(3) = 9 - 18 + 9 = 0$, $(x-3)$ is a factor.

Flashcard 80: What is the product of the zeros of $ax^2+bx+c$ (counting multiplicity)?

Answer: $r_1r_2=\frac{c}{a}$. Vieta's formula relates coefficients to products of roots.

Flashcard 81: Find $f(0)$ for $f(x)=3x^4-5x+12$.

Answer: $f(0)=12$. Substitute $x = 0$ to get the constant term.

Flashcard 82: What is the standard form of a polynomial?

Answer: Terms are ordered by descending powers of $x$. Powers decrease from left to right.

Flashcard 83: State the term for a polynomial with two terms.

Answer: Binomial. A polynomial with exactly two terms.

Flashcard 84: What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n>0$?

Answer: As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient goes from bottom-left to top-right.

Flashcard 85: What is the sum of $4x^2 + 3x + 1$ and $x^2 - x$?

Answer: $5x^2 + 2x + 1$. Add like terms: $(4+1)x^2 + (3-1)x + 1$.

Flashcard 86: Find $f(0)$ for $f(x)=3x^4-5x+12$.

Answer: $f(0)=12$. Substitute $x = 0$ to get the constant term.

Flashcard 87: What is the term for a polynomial with three terms?

Answer: Trinomial. A polynomial with exactly three terms.

Flashcard 88: Identify the zeroes of $x^2 - 5x + 6$.

Answer: $x = 2, x = 3$. Set $(x-2)(x-3) = 0$ and solve.

Flashcard 89: What is the term for a polynomial with three terms?

Answer: Trinomial. A polynomial with exactly three terms.

Flashcard 90: Factor $x^2 + 5x + 6$ completely.

Answer: $(x + 2)(x + 3)$. Find factors of 6 that sum to 5.

Flashcard 91: What is the value of $f(0)$ for $f(x)=a_nx^n+\cdots+a_0$?

Answer: $f(0)=a_0$. Substituting $x = 0$ eliminates all terms except the constant.

Flashcard 92: What is the sum of $x^2 - 3x + 2$ and $x^2 + x - 4$?

Answer: $2x^2 - 2x - 2$. Add corresponding terms.

Flashcard 93: What is the axis of symmetry of $f(x)=ax^2+bx+c$?

Answer: $x=-\frac{b}{2a}$. The axis of symmetry passes through the vertex of the parabola.

Flashcard 94: Identify whether the graph crosses or touches at $x=1$ for $f(x)=(x-1)^4(x+2)$.

Answer: Touches and turns at $x=1$. Even multiplicity $(x-1)^4$ means the graph touches and turns.

Flashcard 95: What is the degree of the polynomial $5x^2y^3 + 3xy - 4$?

Answer: The degree is 5. The sum of exponents in $x^2y^3$ is $2+3=5$.

Flashcard 96: What is the product of $x + 2$ and $x - 3$?

Answer: $x^2 - x - 6$. Use FOIL: $x^2 + 2x - 3x - 6$.

Flashcard 97: What is the constant term in $x^3 + 2x + 5$?

Answer: The constant term is 5. The term without any variable.

Flashcard 98: What is the quadratic formula for solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. The quadratic formula solves $ax^2 + bx + c = 0$ directly.

Flashcard 99: What is the multiplicity rule for a factor $(x-r)^k$ of $f(x)$?

Answer: Zero $r$ has multiplicity $k$. The exponent of the factor equals the multiplicity of the zero.

Flashcard 100: What is the Rational Root Theorem for possible rational zeros of $f(x)$?

Answer: $\frac{p}{q}$ with $p\mid a_0$ and $q\mid a_n$. Rational zeros must have numerator dividing constant term, denominator dividing leading coefficient.