Polynomial Basics
What is a polynomial?
A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable).
How to Add a Polynomial
Example of Addition:
(3x² + 2x -4) + (-2x² + 5x +7)
Align the like terms:
(3x² + (-2x²)) + (2x + 5x) + (-4 +7)
Perform the Addition:
x² + 7x + 3
How to Subtract a Polynomial?
Example of subtraction:
(4x³ + 2x² -5x) – (-3x³ + x² + 7x)
Align the like terms:
(4x³ - (-3x)) + (2x² - x²) + (-5x – 7x)
Perform the subtraction:
7x³ + x² - 12x
If you have 3x - 2 apples and you buy 5x²+ 2x - 1 more apples, how many apples do you have in total?
Answer: You have (3x - 2) + (5x²+ 2x - 1) =
5x²+ 5x - 3 apples in total.
Define the degree of a polynomial.
The degree of a polynomial is the highest exponent among all its terms.
A polynomial with a degree of 0 is a constant (e.g., 5 or −2).
A polynomial with a degree of 1 is a linear polynomial (e.g., x + 1).
A polynomial with a degree of 2 is a quadratic polynomial (e.g., 2x² − 3x + 4).
A polynomial with a degree of 3 is a cubic polynomial (e.g., x³−2x² + 5x − 1).
Add the following polynomials: 3x²+ 2x - 5 and x²- 3x + 1.
Answer: 4x²- x – 4
Find the result of (3y²+ 2y + 1) - (y²- 3y - 2).
Answer: 2y²+ 5y + 3
Mary has 2y²- 3y + 4 books, and John has y²+ 2y - 1 books. How many more books does Mary have than John?
Answer: Mary has (2y²- 3y + 4) - (y²+ 2y - 1) = y²- 5y + 5 more books than John.
What is coefficients in a polynomial?
Coefficients are the numerical values that multiply the variables in each term. They can be positive, negative, or zero.
Example:
2x² - 3x + 4,'
The coefficients are 2, -3, and 4.
What is the result of (4x³+ 2x²+ x) + (3x³- 2x²- 4x)?
Answer: 7x³+ x - 4x
Subtract 4a²+ 5a - 2 from 6a²- 3a + 1.
Answer: 2a²- 8a + 3
A rectangle has a length represented by 3x²- 2x + 1 and a width represented by x²+ 3x - 2. What is the polynomial expression for its area?
Answer: The area of the rectangle is given by (3x²- 2x + 1)(x²+ 3x - 2).
Explain the difference between a monomial and a binomial.
Perform the following polynomial addition: (a²+ 2a + 1) + (2a²- 3a + 4).
Answer: 3a²- a + 5
What is the result of (2x³+ 4x²- x + 3) - (x³- 2x²+ 4x - 1)?
Answer: x³+ 6x²- 5x + 4
Jay has chocolate, and Elle has y - 1 books. How many more books does Jay have than Elle?
Answer: (5y - 3) + (y-1) = 6y - 4
Give an example of a polynomial in standard form.
P(x)=Ax4−Bx3+Cx2−Dx+1
Add the polynomials: 5x³+ 2x²- 7x + 1 and -2x³+ 3x²- 1x - 4.
Answer: 3x³+ 5x²- 8x - 3
Subtract -3x ⁴+ 2x³- 5x²+ 4x - 1 from 4x ⁴- x³ + 2x²- 3x + 5.
Answer: 7x⁴- 3x³+ 7x²- 7x + 6
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