1. Write a polynomial in standard form that has solutions: 0, -2, 3:
x(x+2)(x-3)
Expand this to write it in standard form.
2. Write a trinomial that has a degree of 4 and a lead coefficient of -3
3. True of false: 3/x^2 is a polynomial expression
x(x+2)(x-3)
Expand this to write it in standard form.
x(x+2)(x-3) is going to have an x^3 in it for starters
Multiply that all out more carefully.
x (x^2 - x - 6) =
x^3 - x^2 - 6 x
write an expression with three terms
the first term, with the highest power of x has a coef of -3
That highest power of x is 4
the other two terms can have x^3, x^2, x^1 (which is x) and x^0 (which is one)
To find the polynomial, we multiply these factors together:
(x)(x + 2)(x - 3)
Expanding this expression, we get:
(x^2 + 2x)(x - 3)
Now, we can multiply each term by x:
x^3 - 3x^2 + 2x^2 - 6x
Combining like terms, we have:
x^3 - x^2 - 6x
Thus, the polynomial in standard form with the given solutions is x^3 - x^2 - 6x.
2. To write a trinomial with a degree of 4 and a lead coefficient of -3, we can start by using the generic form of a polynomial:
ax^4 + bx^3 + cx^2 + dx + e
Given that the lead coefficient is -3, we have:
-3x^4 + bx^3 + cx^2 + dx + e
Since the degree of the trinomial is 4, it means that the exponent of the leading term is 4. Therefore, the coefficient of x^4 is -3.
Thus, a trinomial with a degree of 4 and a lead coefficient of -3 can be represented as -3x^4 + bx^3 + cx^2 + dx + e.
3. False. The expression 3/x^2 is not a polynomial because it has a variable in the denominator. Polynomial expressions must have non-negative integer exponents on variables. In this case, the expression has x^(-2), which is equivalent to 1/x^2. Therefore, the expression 3/x^2 is considered a rational function, not a polynomial.