1)Expand and simplify. Express each equation in standard form.
a) f(x) = -2(x-1)(x+4)(x+1)(x-3) b) f(x) = (x+2)(x+1)(x+4)
c) f(x) = x(x-6)
2) For each function in question 1. state the degree of the polynomial and identify the type of function
1a. F(x) = -2(x^2-1)(x+4)(x-3).
F(x) = -2(x^2-1)(x^2-3x+4x-12).
F(x) = -2(x^2-1)(x^2+x-12).
F(x) = (-2x^2+2)(x^2+x-12).
F(x) = -2x^4-2x^3+24x^2.
b. F(x) = (x+2)(x+1)(x+4).
F(x) = (x^2+x+2x+2)(x+4).
F(x) = x^3+4x^2+x^2+4x+2x^2+8x+2x+8.
F(x) = x^3+7x^2+14x+8.
c. F(x) = x(x-6).
F(x) = x^2-6x.
2a. 4th degree.
b. 3rd degree.
c. 2nd degree.
X²-10x=5
f(x) = (x+1)(x+3)(x-5)
To expand and simplify the given equations, we need to multiply and combine like terms. Let's go through each equation step by step:
a) f(x) = -2(x-1)(x+4)(x+1)(x-3)
To expand this equation, we can use the distributive property to multiply each term inside the parentheses.
f(x) = -2(x^2 + 4x - x + 4)(x-3)
Simplifying further:
f(x) = -2(x^2 + 3x + 4)(x-3)
Using the distributive property again:
f(x) = -2x^3 - 6x^2 - 8x + 6x^2 + 18x + 24
Combining like terms:
f(x) = -2x^3 + 12x + 24
The equation is now in expanded form. To express it in standard form, we arrange the terms in descending order of degree:
f(x) = -2x^3 + 12x + 24 (standard form)
The degree of this polynomial function is 3.
b) f(x) = (x+2)(x+1)(x+4)
Using the distributive property and multiplying each term inside the parentheses, we get:
f(x) = (x^2 + 3x + 2)(x+4)
Expanding further:
f(x) = x^3 + 4x^2 + 3x^2 + 12x + 2x + 8
Combining like terms:
f(x) = x^3 + 7x^2 + 14x + 8
The equation is now in expanded form. To express it in standard form, let's arrange the terms:
f(x) = x^3 + 7x^2 + 14x + 8 (standard form)
The degree of this polynomial function is 3.
c) f(x) = x(x-6)
Using the distributive property:
f(x) = x^2 - 6x
The equation is already in expanded form. To express it in standard form, we arrange the terms:
f(x) = x^2 - 6x (standard form)
The degree of this polynomial function is 2.
In summary:
a) f(x) = -2x^3 + 12x + 24 (standard form), degree: 3, type: cubic function
b) f(x) = x^3 + 7x^2 + 14x + 8 (standard form), degree: 3, type: cubic function
c) f(x) = x^2 - 6x (standard form), degree: 2, type: quadratic function