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)⋅(bm xm+⋯+b1 x+b0)(an xn+⋯+a1 x+a0)⋅(bm xm+⋯+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).
Read More: Polynomials
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.