Algebra 2 – Polynomial Functions – Flashcards

a real number, variable or product of a real number and one or more variables with whole-number exponents ex. 4x⁴yz²

two monomials connected with addition or subtraction ex. 3x²+5xy⁴ or 5x⁴-3x³

three monomials connected with addition or subtraction ex. x⁴y⁶+x³y⁵+x²z

a monomial or a sum of monomials

the value of the greatest exponent for a polynomial in one variable

an equation with a polynomial

a polynomial arranged by degree in decending numerical order P(x) = 4x³+3x²+5x-2

place where the graph of a polynomial changes direction

direction of the graph to the far left and the far right

the expression x−a is a factor of a polynomial if and only if a is a zero of the related polynomial function

the zero of a repeated linear factor

the exponent of a repeated linear factor

the value of a function at an up-to-down turning point

the value of a function at a down-to-up turning point

a³+b³=(a+b)(a²-ab+b²)

a³-b³=(a-b)(a²+ab+b²)

a²+2ab+b²=(a+b)² a²-2ab+b²=(a-b)²

a²-b²=(a-b)(a+b)

ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)

complex roots occur in conjugate pairs: →if a+√b is a root of the polynomial, then a-√b is also a root of the polynomial (√ is square root) →if a+bi is a root of the polynomial, then a-bi is also a root of the polynomial →by the same rule: if bi (or √b) is a root so is -bi (or -√b)

allows one to determine the possible number of positive or negative real roots a polynomial, written in standard form, has: Positive Real Roots → equal to the number of sign changes between consecutive coefficients of the polynomial, P(x), or less than that by an even number Negative Real Roots → equal to the number of sign changes between consecutive coefficients of the changed polynomial, P(-x), or less than that by an even numbermultiple roots are still counted by their multiplicity

There are a limited number of rational roots of any polynomial; the possible rational roots can be obtained by finding all fractions that fit the following criteria: the numerator of the fraction is a factor of the constant term of polynomial & the denominator of the fraction is a factor of the leading coefficient of the polynomial.The fraction can be positive or negative and must always be in reduced form.

a quick way to find the remainder of a polynomial long-division problem. Given P(x) ÷ (x-a); the remainder is P(a) → substitute "a" into the equation for "x" and simplify to find the remainder of the polynomial.

makes polynomial division easier when dividing by a linear factor (x-a) or (x+a) →write the coefficients, including zeros, of the polynomial in standard form (leave out the variables and exponents) →divide by the opposite of "a" →"drop down" the first coefficient →multiply by the divisor and add to the next coefficient →repeat the previous step until there are no more coefficients to add