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Value Of A Polynomial

The value of a polynomial is the result obtained when the variable in the polynomial expression is replaced by a specific real number. If p(x) is a polynomial in x, and k is any real number, then the value obtained by replacing x by k in p(x) is called the value of p(x) at x = k, denoted by p(k).

Key Concepts

To calculate the value of a polynomial, substitute the given real number for every instance of the variable in the algebraic expression.
After substitution, simplify the mathematical expression using the standard order of operations to find a single numerical value.
If the calculated value of the polynomial at a specific point k is zero, then k is officially called a zero or root of the polynomial.

Formula & Equation

p(k)=an(k)n+anāˆ’1(k)nāˆ’1+...+a1(k)+a0p(k) = a_n(k)^n + a_{n-1}(k)^{n-1} + ... + a_1(k) + a_0

p(k) is the value of the polynomial, k is the substituted real number, and a_n to a_0 are the real coefficients of the polynomial terms.

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Common Misconceptions

Myth: Students often confuse the value of a polynomial with the zero of a polynomial, thinking they are the same concept.
Fact: The value of a polynomial can be any real number obtained after substitution. A zero of a polynomial is a very specific input value that makes the final output exactly zero.

Real World Applications

If the profit of a business is modeled by the quadratic polynomial p(x) = 5x^2 - 3x + 10 where x is the number of items sold, finding p(10) gives the exact profit for 10 items.
If the distance traveled by a vehicle is given by the polynomial d(t) = 4t^2 + 2t, substituting t = 3 seconds gives the exact distance covered at that specific moment.

Frequently Asked Questions

How do you find the value of a polynomial at a given point?
You find it by replacing the variable in the polynomial with the given real number and performing the arithmetic operations to simplify the expression.
What happens if the value of a polynomial is zero?
If the value of a polynomial p(x) at x = k is exactly zero, then the real number k is known as the zero of the polynomial.

Keep Learning

PolynomialZero of a PolynomialReal Number