MP Board · Class 10 · Mathematics · PolynomialsExplain the concept of polynomials and their types. Also, find the value of the polynomial $f(x) = 2x^3 - 5x^2 + x - 1$ when $x = 2$. Provide a step-by-step solution for the numerical part.

Introduction to Polynomials\nPolynomials are algebraic expressions that consist of variables and coefficients combined using only addition, subtraction, and multiplication. They are an essential part of algebra and are used to solve a wide range of problems in mathematics, science, and engineering.

Types of Polynomials

• Monomials: These are polynomials with only one term. Examples include $3x$, $4y^2$, and $2z^3$.
• Binomials: These are polynomials with two terms. Examples include $x + y$ and $3x^2 - 2y$.
• Trinomials: These are polynomials with three terms. An example is $x^2 + 2x + 1$.

Evaluating Polynomials\nTo evaluate a polynomial at a given value of $x$, we substitute $x$ in the polynomial and perform the operations.

Evaluating $f(x) = 2x^3 - 5x^2 + x - 1$ at $x = 2$

1. Substitute $x = 2$ into the polynomial: $f(2) = 2(2)^3 - 5(2)^2 + 2 - 1$
2. Calculate the powers of $2$: $(2)^3 = 8$ and $(2)^2 = 4$
3. Substitute these values back into the equation: $f(2) = 2(8) - 5(4) + 2 - 1$
4. Perform the multiplication: $f(2) = 16 - 20 + 2 - 1$
5. Finally, perform the addition and subtraction: $f(2) = 16 - 20 + 2 - 1 = -3$ \nThe value of the polynomial $f(x) = 2x^3 - 5x^2 + x - 1$ when $x = 2$ is $-3$.