Polynomials are fundamental algebraic expressions that consist of variables and coefficients, incorporating the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding polynomials is crucial for solving various mathematical problems in algebra and calculus.

What are Polynomials?

A polynomial is an expression of the form:

P(x)=aⁿ xⁿ+aⁿ⁻¹ xⁿ⁻¹+⋯+a¹x+a⁰

where aⁿ,aⁿ⁻¹,…,a¹,a⁰​ are constants (coefficients) and x is a variable. The degree of the polynomial is the highest power of x that appears in the polynomial.

Important Formulas Related to Polynomials

Sum of Polynomials:

(aⁿ xⁿ+⋯+a¹ x+a⁰)+(bⁿ xⁿ+⋯+b¹ x+b⁰)=(aⁿ+bⁿ)xⁿ+⋯+(a¹+b¹)x+(a⁰+b⁰)

The sum of two polynomials P(x) and Q(x) is defined as the polynomial (P(x) + Q(x))(x), where each term of P(x) is added to the corresponding term of Q(x) with the same exponent.

Product of Polynomials:

(aⁿ xⁿ+⋯+a¹ x+a⁰)⋅(b^m x^m+⋯+b¹ x+b⁰)(aⁿ ​xⁿ+⋯+a¹ ​x+a⁰​)⋅(b^m ​x^m+⋯+b¹ ​x+b⁰​)

The product of two polynomials P(x) and Q(x) is defined as the polynomial (P(x) × Q(x))(x), where each term of P(x) is multiplied by the corresponding term of Q(x).

Derivative of a Polynomial:

d/dx(aⁿ xⁿ+⋯+a¹ x+a⁰)=naⁿ xⁿ⁻¹+⋯+a¹

The derivative of a polynomial function P(x) with respect to x is another polynomial function denoted as P′(x). The derivative of P(x) with respect to x can be calculated using the formula P′(x) = (dP(x) / dx). The formula for calculating the derivative of a term with an exponent in a polynomial is (dP(x) / dx) = (coefficient × exponent). When the exponent is negative, the coefficient is negative and the exponent becomes positive (exponent – 1).

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Solved Questions on Polynomials

Problem 1: Find the product of P(x)=x³−2x²+x−4 and Q(x)=2x+3

Solution:

P(x)⋅Q(x)=(x³−2x²+x−4)(2x+3)

=x³(2x+3)−2x²(2x+3)+x(2x+3)−4(2x+3)

=2x⁴+3x³−4x³−6x²+2x²+3x−8x−12

=2x⁴−x³−4x²−5x−12

Problem 2: Determine the sum of P(x)=5x²−3x+1 and Q(x)=−2x²+4x−6.

Solution:

P(x)+Q(x)=(5x²−3x+1)+(−2x²+4x−6)

=5x²−2x²−3x+4x+1−6

=3x²+x−5

Problem 3: Compute the derivative of P(x)=3x⁴−5x²+6x+8.

Solution:

P′(x)=d/dx(3x⁴−5x²+6x+8)

=12x³−10x+6

Problem 4: If P(x)=x³+x²−x+1 and Q(x)=2x³−3x+4, find P(x)−Q(x).

Solution:

P(x)−Q(x)=(x³+x²−x+1)−(2x³−3x+4)

=x³−2x³+x²−x+3x+1−4

=−x³+x²+2x−3

Problem 5: Evaluate P(x) at x=2 for P(x)=4x³−3x+5.

Solution:

P(2)=4(2)3−3(2)+5

=4⋅8−6+5

=32−6+5=31

Problem 6: Find the quotient and remainder when P(x)=x⁴−2x³+3x²−x+6 is divided by x−1.

Solution:

Using synthetic division:

Quotient: x3−x2+2x+1

Remainder: 7

Problem 7: If P(x)=x²+2x+1 and Q(x)=x+1, find P(x)/Q(x).

Solution:

P(x)/Q(x)=x²+2x+1 / x+1=(x+1)²/x+1 = x+1 (x≠−1)

Problem 8: Compute the integral of the polynomial P(x)=x³−2x²+x−4.

Solution:

∫P(x) dx=∫(x³−2x²+x−4) dx

=x⁴/4−2x³/3+x²/2−4x+C

Practice Questions on Polynomials

1. Factor the polynomial 2x³−5x²+3x.

2. Find all real solutions of the equation x⁴−16=0

3. Simplify the expression (3x²−4x+1)(x²+2x−8).

4. Determine the degree and leading coefficient of the polynomial −4x⁵+2x³−7x+1.

5. Find the sum of the coefficients of the polynomial 4x³−2x²+5x−1.

6. Factor completely the polynomial x⁴−5x²+4.

7. Solve the inequality x³−9x≥0.

8. Determine if the polynomial x³−2x²+4x−8 has any real zeros.

9. Given that x−2 is a factor of 2x³−7x²+3x+6, find the remaining factor.

10. Evaluate the polynomial 3x²−2x+5 when x=2.

Polynomials- FAQs

What is a polynomial?

Polynomials are fundamental algebraic expressions that consist of variables and coefficients, incorporating the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

How do you find the degree of a polynomial?

The degree of a polynomial is the highest power of its variable.

How do you add polynomials?

(aⁿ xⁿ+⋯+a¹ x+a⁰)+(bⁿ xⁿ+⋯+b¹ x+b⁰)=(aⁿ+bⁿ)xⁿ+⋯+(a¹+b¹)x+(a⁰+b⁰)

The sum of two polynomials P(x) and Q(x) is defined as the polynomial (P(x) + Q(x))(x), where each term of P(x) is added to the corresponding term of Q(x) with the same exponent.