zero
(noun)
Also known as a root; an
(noun)
Also known as a root, a zero is an
(noun)
Also known as a root; an
(noun)
Also known as a root; an x value at which the function of x is equal to zero.
(noun)
Also known as a root, an
(noun)
Also known as a root, a zero is an
(noun)
Also known as a root, a zero is an
Examples of zero in the following topics:
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Zeroes of Linear Functions
- A zero, or $x$-intercept, is the point at which a linear function's value will equal zero.
- Zeros can be observed graphically.
- Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.
- The zero is $(-4,0)$.
- The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.
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Finding Polynomials with Given Zeros
- To construct a polynomial from given zeros, set $x$ equal to each zero, move everything to one side, then multiply each resulting equation.
- One type of problem is to generate a polynomial from given zeros.
- If it is not specified what the multiplicity of the zeros are, we want the zeros to have multiplicity one.
- There are no other zeros, i.e. if a number is not mentioned in the problem statement, it cannot be a zero of the polynomial we find.
- Two polynomials with the same zeros: Both $f(x)$ and $g(x)$ have zeros $0, 1$ and $2$.
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Finding Zeros of Factored Polynomials
- An $x$ -value at which this occurs is called a "zero" or "root. "
- A polynomial function may have many, one, or no zeros.
- All polynomial functions of positive, odd order have at least one zero (this follows from the fundamental theorem of algebra), while polynomial functions of positive, even order may not have a zero (for example $x^4+1$ has no real zero, although it does have complex ones).
- Replacing $x$ with a value that will make either $(x+3),(x+1)$ or $(x-2)$ zero will result in $f(x)$ being equal to zero.
- Use the factored form of a polynomial to find its zeros
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Rational Inequalities
- The zeros in the denominator are $x$-values are at which the rational inequality is undefined, the result of dividing by zero.
- The numerator has zeros at $x=-3$ and $x=1$.
- The denominator has zeros at $x=-2$ and $x=2$.
- For $x$ values that are zeros for the numerator polynomial, the rational function overall is equal to zero.
- Solve for the zeros of a rational inequality to find its solution
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Polynomial Inequalities
- The best way to solve a polynomial inequality is to find its zeros.
- The three terms reveal zeros at $x=-3$, $x=-1$, and $x=2$.
- The next zero is at $x=-1$.
- At $x=-1$, $(x+1)$ equals zero, becoming positive to the right.
- Solve for the zeros of a polynomial inequality to find its solution
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Linear Equations in Standard Form
- where $A$ and $B$ are both not equal to zero.
- Recall that a zero is a point at which a function's value will be equal to zero ($y=0$), and is the $x$-intercept of the function.
- Therefore, the zero of the equation occurs at $x = \frac{5}{1} = 5$.
- The zero is the point $(5, 0)$.
- Example: Find the zero of the equation $3(y - 2) = \frac{1}{4}x +3$
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Zeroes of Polynomial Functions With Rational Coefficients
- In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
- Zero divided by any other integer equals zero.
- Therefore zero is a rational number, but division by zero is undefined.
- By setting each term to zero, it can be found that the zeros for this equation are x=-6 and x=-9/2.
- Extend the techniques of finding zeros to polynomials with rational coefficients
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The Discriminant
- That is, it is the $x$-coordinate at which the function's value equals zero.
- If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
- Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
- Because Δ is greater than zero, the function has two distinct, real roots.
- Because the value is greater than 0, the function has two distinct, real zeros.
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Zeros of Polynomial Functions with Real Coefficients
- A root, or zero, of a polynomial function is a value that can be plugged into the function and yield $0$.
- The zero of a function, $f(x)$, refers to the value or values of $x$ that will result in the function equaling zero, $f(x)=0$.
- Use the quadratic equation and factoring methods to find the zeros of a polynomial
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Basics of Graphing Polynomial Functions
- Between two zeros (and before the smallest zero, and after the greatest zero) a function will always be either positive, or negative.
- Every time we cross a zero of odd multiplicity (if the number of zeros equals the degree of the polynomial, all zeros have multiplicity one and thus odd multiplicity) we change sign.
- The graph of the zero polynomial $f(x)=0$ is the $x$-axis, since all real numbers are zeros.
- It has exactly 6 zeroes and 5 local extrema.
- It has 3 real zeros (and two complex ones).