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Let $f(x)$ be a quadratic polynomial such that $f(-4) = -22,$ $f(-1)=2$, and $f(2)=-1.$ Let $g(x) = f(x)^{16}.$ Find the sum of the coefficients of the terms in $g(x)$ that have even degree. (For example, the sum of the coefficients of the terms in $-7x^3 + 4x^2 + 10x - 5$ that have even degree is $(4) + (-5) = -1.$)
Answers
Some polynomials with real coefficients, like
x
2
+
1
, have no real zeros. As it turns out, every polynomial with a complex coefficient has a complex zero. Every polynomial of odd degree with real coefficients has a real zero.
The fundamental theorem states that every non-constant, single-variable polynomial with complex coefficients has at least one complex root.
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Answer:
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Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2
Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2will give the expression $x^{2}$ a name (or a variable in the computer science sense) called f. By “doing” something to f, we are in fact doing that to $x^{2}$. A better example is if we type
Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2will give the expression $x^{2}$ a name (or a variable in the computer science sense) called f. By “doing” something to f, we are in fact doing that to $x^{2}$. A better example is if we typeg:=(x+2)^2
Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2will give the expression $x^{2}$ a name (or a variable in the computer science sense) called f. By “doing” something to f, we are in fact doing that to $x^{2}$. A better example is if we typeg:=(x+2)^2and the we can expand it by
Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2will give the expression $x^{2}$ a name (or a variable in the computer science sense) called f. By “doing” something to f, we are in fact doing that to $x^{2}$. A better example is if we typeg:=(x+2)^2and the we can expand it byexpand(g)
Below we will be plotting $x^{2}$ with many different options. In the case of $x^{2}$ it is short enough to type out each time, but if we have a complicated function, we probably don’t want to. In such a case, we will store the expression as a name. To do this, we’ll use :=. For examplef:=x^2will give the expression $x^{2}$ a name (or a variable in the computer science sense) called f. By “doing” something to f, we are in fact doing that to $x^{2}$. A better example is if we typeg:=(x+2)^2and the we can expand it byexpand(g)which returns $x^{2}+4x+4$.