Polynomials
60 flashcards covering Polynomials for the ACT Math section.
Polynomials are mathematical expressions made up of variables and coefficients, combined with addition, subtraction, and multiplication by non-negative powers. For instance, something like 2x^3 + 4x^2 - 5x + 1 is a polynomial, where the variables are raised to whole number exponents. They form the building blocks of algebra, helping to describe relationships in equations and model real-world scenarios, such as motion or growth patterns.
On the ACT Math section, polynomials show up in questions that involve factoring, simplifying expressions, solving equations, or graphing functions. Common traps include errors in distributing negative signs, overlooking extraneous solutions, or confusing polynomial degrees with their roots. Focus on mastering key skills like factoring techniques, the quadratic formula, and identifying intercepts to handle these problems efficiently.
Practice factoring quadratics regularly to spot patterns quickly.
Terms (60)
- 01
Polynomial
A polynomial is an algebraic expression that consists of variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Monomial
A monomial is a polynomial with only one term, such as 3x^2 or 5.
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Binomial
A binomial is a polynomial with exactly two terms, like x + 2 or 3y - 4.
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Trinomial
A trinomial is a polynomial with exactly three terms, such as x^2 + 2x + 1.
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Degree of a polynomial
The degree of a polynomial is the highest exponent of the variable in the expression, which determines its behavior and complexity.
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Leading coefficient
The leading coefficient of a polynomial is the coefficient of the term with the highest degree, affecting the graph's scale and direction.
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Constant term
The constant term in a polynomial is the term without a variable, representing the value when the variable is zero.
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Standard form of a polynomial
Standard form of a polynomial arranges terms in descending order of their degrees, making it easier to identify the degree and leading coefficient.
- 09
Adding polynomials
Adding polynomials involves combining like terms from each polynomial, which means adding the coefficients of terms with the same variables and exponents.
- 10
Subtracting polynomials
Subtracting polynomials requires distributing the negative sign and then combining like terms, similar to addition but accounting for the subtraction.
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Multiplying polynomials
Multiplying polynomials means using the distributive property to multiply each term in one polynomial by each term in the other and then combining like terms.
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FOIL method
The FOIL method is a technique for multiplying two binomials by multiplying the First terms, Outer terms, Inner terms, and Last terms, then combining like terms.
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Distributive property in polynomials
The distributive property in polynomials allows multiplying a monomial by each term inside parentheses, such as distributing 2x to (x + 3) to get 2x^2 + 6x.
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Factoring a polynomial
Factoring a polynomial means rewriting it as a product of simpler polynomials, often to solve equations or simplify expressions.
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Greatest common factor (GCF)
The greatest common factor of a polynomial is the largest polynomial that divides evenly into each term, which is factored out first in many problems.
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Factoring quadratics
Factoring quadratics involves rewriting a quadratic expression as a product of two binomials, such as turning x^2 + 5x + 6 into (x + 2)(x + 3).
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Difference of squares
The difference of squares is a factoring pattern for expressions like a^2 - b^2, which factors into (a - b)(a + b).
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Perfect square trinomial
A perfect square trinomial is an expression like (a + b)^2, which expands to a^2 + 2ab + b^2 and can be factored back accordingly.
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Sum of cubes
The sum of cubes is a factoring formula for expressions like a^3 + b^3, which factors into (a + b)(a^2 - ab + b^2).
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Difference of cubes
The difference of cubes is a factoring formula for expressions like a^3 - b^3, which factors into (a - b)(a^2 + ab + b^2).
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Factoring by grouping
Factoring by grouping involves rearranging and factoring common factors from pairs of terms in a polynomial with four or more terms.
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Quadratic formula
The quadratic formula is a method to solve ax^2 + bx + c = 0, giving roots as x = [-b ± sqrt(b^2 - 4ac)] / (2a).
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Discriminant
The discriminant of a quadratic equation ax^2 + bx + c = 0 is b^2 - 4ac, which indicates the nature of the roots: positive for two real roots, zero for one real root, and negative for no real roots.
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Roots of a polynomial
Roots of a polynomial are the values of the variable that make the polynomial equal zero, representing where the graph crosses the x-axis.
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Zero of a function
A zero of a function is a value of the input that makes the output zero, equivalent to the roots of the polynomial.
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Multiplicity of a root
The multiplicity of a root is the number of times it appears as a factor in the polynomial, affecting how the graph touches or crosses the x-axis.
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End behavior of polynomials
End behavior describes how the graph of a polynomial function behaves as x approaches positive or negative infinity, determined by the degree and leading coefficient.
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Synthetic division
Synthetic division is a shorthand method for dividing a polynomial by a linear factor like (x - c), using only the coefficients to find the quotient and remainder.
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Remainder theorem
The remainder theorem states that the remainder of a polynomial f(x) divided by (x - c) is f(c), allowing quick evaluation without full division.
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Factor theorem
The factor theorem states that (x - c) is a factor of a polynomial f(x) if and only if f(c) = 0.
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Rational root theorem
The rational root theorem provides possible rational roots of a polynomial with integer coefficients by suggesting factors of the constant term divided by factors of the leading coefficient.
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Descartes' rule of signs
Descartes' rule of signs estimates the possible number of positive and negative real roots by counting sign changes in f(x) and f(-x).
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Polynomial long division
Polynomial long division is a process similar to numerical long division for dividing one polynomial by another, resulting in a quotient and remainder.
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Binomial theorem
The binomial theorem provides a formula to expand (a + b)^n as a sum of terms, calculated using combinations and powers.
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Pascal's triangle
Pascal's triangle is a triangular array of numbers used to find the coefficients in the expansion of binomials, with each number being the sum of the two above it.
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Evaluating polynomials
Evaluating a polynomial means substituting a specific value for the variable to find the output, often used to check roots or behavior.
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Graphing polynomials
Graphing polynomials involves plotting points, identifying roots, and considering end behavior and turning points to sketch the curve.
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Intercepts of a polynomial
Intercepts of a polynomial are the points where the graph crosses the axes, with x-intercepts at the roots and y-intercept at the value when x is zero.
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Vertex form of a quadratic
Vertex form of a quadratic is y = a(x - h)^2 + k, where (h, k) is the vertex, making it easier to graph and identify the maximum or minimum.
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Factored form of a polynomial
Factored form expresses a polynomial as a product of its factors, such as y = a(x - r1)(x - r2), highlighting the roots.
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Completing the square
Completing the square rewrites a quadratic equation in vertex form by adding and subtracting a constant, useful for graphing or solving.
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Solving polynomial inequalities
Solving polynomial inequalities involves finding the roots, testing intervals on a number line, and determining where the inequality holds based on the graph.
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Common mistakes in factoring
Common mistakes in factoring include forgetting to factor out the GCF first or incorrectly applying patterns like difference of squares.
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Expanding (x + 2)^3
Expanding (x + 2)^3 using the binomial theorem gives x^3 + 3x^2(2) + 3x(2)^2 + (2)^3, which simplifies to x^3 + 6x^2 + 12x + 8.
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Factoring x^2 + 5x + 6
Factoring x^2 + 5x + 6 involves finding two numbers that multiply to 6 and add to 5, resulting in (x + 2)(x + 3).
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Strategy for solving cubic equations
A strategy for solving cubic equations is to use the rational root theorem to test possible roots, then factor and solve the resulting quadratic.
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Identifying even degree polynomials
Even degree polynomials have an even highest exponent, leading to end behavior where both ends go in the same direction.
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Determining number of turning points
The number of turning points in a polynomial graph is at most one less than its degree, indicating changes in direction.
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Relative maximum and minimum
Relative maximum and minimum are the highest and lowest points on a polynomial graph in a local region, often found at turning points.
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Polynomial functions vs. equations
Polynomial functions express y as a polynomial in x, while equations set them equal to zero, requiring solutions for the variable.
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Coefficient of a term
The coefficient of a term in a polynomial is the numerical factor multiplying the variable, such as 3 in 3x^2.
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Like terms in polynomials
Like terms in polynomials are terms with the same variables raised to the same powers, which can be combined by adding or subtracting their coefficients.
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Simplifying polynomials
Simplifying polynomials means combining like terms and factoring if possible to write the expression in its most basic form.
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Expanding binomials
Expanding binomials involves multiplying the terms, such as (a + b)(c + d) to get ac + ad + bc + bd.
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Multiplying binomials
Multiplying binomials uses methods like FOIL for two terms or distribution for more, resulting in a polynomial with combined like terms.
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Dividing polynomials by monomials
Dividing polynomials by monomials involves dividing each term in the polynomial by the monomial, simplifying coefficients and exponents accordingly.
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Polynomial equations
Polynomial equations set a polynomial equal to zero, requiring the use of factoring or formulas to find the values of the variable that satisfy it.
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Systems of polynomial equations
Systems of polynomial equations involve solving multiple equations simultaneously, often by substitution or elimination, to find common solutions.
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Applications of polynomials
Applications of polynomials include modeling real-world scenarios like area or volume problems, where expressions represent quantities that vary.
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Word problems involving polynomials
Word problems involving polynomials translate scenarios into polynomial equations, such as finding dimensions that maximize area, and solve them accordingly.