polynomials
decide whether the function is a polynomial function. if so, write it in the standard form and state its degree, type, leading coefficient
f(x)= -3x+5x3-6x2+2
f(x)=5x3-6x2-3x+2
degree:3
type:cubic
leading coefficient: 5
(2x3+6x2−x+ 1)-(8x3− 3x2− 2x+ 9)
6x3− 9x2−x + 8
(x3 - 7x2 + 6x4)+(-6x4 + x3 + 3x2)
2x2-4x2
(y+ 5) (3y2− 2y+ 2)
3y3+ 13y2− 8y+ 10
what is a polynomial
is a monomial or a sum of monomials
decide whether the function is a polynomial function. if so, write it in the standard form and state its degree, type, leading coefficient
f(x)= 9x4+8x3-6x2+2x
not a polynomial
(3z2+z− 4)-(2z2+ 3z)
−z2+ 2z+ 4
(n5-6-7n4)+(8n5-n4-6n3)
9n5-8n4-6n3-6
(−x2+ 2x+ 4) (x− 3)
3x2− 6x − 12
end behavior
a function’s graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (−∞).
evaluate the function for the given value of x
h(x)= -3x4+2x3-12x-6;x=-2
h(x)=-46
(3x3-2x2+4x-8)-(5x3+12x2-3x-4)
-2x3-14x2+7x-4
(4a4-2a3+7a2)+(-6a2-2a3+5a4)
9a4+a2
(x− 1) (x+ 4) (x+ 5)
x3+ 8x2+ 11x − 20
pascals triangle
When you arrange the coefficients of the variables in the expansion of (a+b)n, you will see a special pattern
describe end behavior
h(x)= -5x4+7x3-6x2+9x+2
the function has an even degree and a negative leading coefficient
f(x)=-∞ as x -∞
f(x)=-∞ as x ∞
(7x4-9x3-4x2+5x+6)-(2x4+3x3-x2+x-4)
5x4-12x3-3x2+4x+10
(-8k+3k3-k4)+(-8k3+8k2-4k)
10m5-3m3-15
(2x+4)(2x2-8x+3)
4x3-8x2-26x+12
polynomial function
is a function of the form f(x)=an xn+an−1xn-1+⋅⋅⋅ +a1x+a0
describe the end behavior
g(x)= 7x7+12x8-6x3-2x-18
the degree is odd and the leading coefficient is positive
g(x)= -∞ as x -∞
g(x)= ∞ as x ∞
(5x6-2x4+9x3+2x-4)-(7x5-8x4+2x-11)
5x6-7x5+6x4+9x3+7
(3x3+ 2x2−x− 7)+(x3− 10x2+ 8 )
4x3− 8x2−x + 1
(3x+4)(7x2+2x-9)
21x3+34x2-19x-36
quadratic function
is a polynomial function with one or more variables in which the highest-degree term is of the second degree