Value of a Polynomial

The value of a polynomial at a particular number is obtained by substituting that number in place of the variable. This process helps understand how the polynomial behaves for different values and is also useful in checking zeros, solving problems, and verifying algebraic identities.

Example: For \(p(x) = x^2 + 3x - 1\), if we want the value at \(x = 2\):

\(p(2) = 2^2 + 3(2) - 1 = 4 + 6 - 1 = 9\)

To find the value of a polynomial at a given number, follow these simple steps:

1. Identify the value to be substituted (e.g., \(x = 3\)).
2. Replace the variable in the polynomial with this value.
3. Perform the arithmetic operations step by step.
4. Write the final simplified answer.

Sometimes we want the value of a polynomial at several points. A value table helps us compare how the polynomial behaves for different values of the variable.

Example: For \(p(x) = x^2 - x\):

Let’s calculate values for various polynomials.

A number is a zero of a polynomial when the value of the polynomial is zero at that number.

Example:

Polynomial: \(p(x) = x^2 - 4\)

Check \(x = 2\):

\(p(2) = 4 - 4 = 0\)

So, 2 is a zero.

• Incorrect substitution of negative numbers.
• Forgetting to apply exponents before multiplication.
• Solving in the wrong order (always follow BODMAS).
• Confusing value of polynomial with zeros or factors.

Find the value of each polynomial at the given value:

1. \(p(x) = 3x - 7\), find \(p(5)\)
2. \(p(x) = x^2 + 2x + 1\), find \(p(-2)\)
3. \(p(t) = t^3\), find \(p(3)\)
4. \(p(y) = y^2 - 5y\), find \(p(0)\)

• Value of a polynomial is found by substituting the number for the variable.
• Follow BODMAS rules while simplifying.
• Value tables help compare how polynomials behave at different inputs.
• A number is a zero of a polynomial if its substitution value is zero.