Polynomials I

What Are Polynomials?

A polynomial is an expression containing constants and variables connected only through basic operations of algebra.

A polynomial is a finite expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and taking non-negative integer powers.
A polynomial can be written as the sum of a finite number of terms. Each term consists of the product of a constant (called the coefficient of the term) and a finite number of variables (usually represented by letters) raised to integer powers.

References

“Polynomials – What Are Polynomials? Definition And Examples”. 2022. CUEMATH. https://www.cuemath.com/algebra/polynomials/.

“Polynomials (Definition, Types and Examples)”. 2022. BYJUS. https://byjus.com/maths/polynomial/.

“Polynomials Intro (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-arithmetic/x2ec2f6f830c9fb89:poly-intro/v/polynomials-intro.

“Polynomials”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/polynomials.html.

Adding and Subtracting Polynomials

Polynomials can be added or subtracted by combining like terms.

The rules for adding and subtracting algebraic expressions apply to polynomials; only like terms can be combined.
Any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials.
The sum or difference of two polynomials will have the same degree as the polynomial with the higher degree in the problem.

References

“6.4 Adding and Subtracting Polynomials”. CK-12 Foundation. 2022. https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/6.4/primary/lesson/adding-and-subtracting-polynomials-alg-ii/.

“9.1 Addition and Subtraction of Polynomials”. 2022. CK-12 Foundation. https://www.ck12.org/book/ck-12-algebra-i-second-edition/section/9.1/.

“Adding And Subtracting Polynomials”. 2022. CUEMATH. https://www.cuemath.com/algebra/adding-and-subtracting-polynomials/.

“Adding And Subtracting Polynomials”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/polynomials-adding-subtracting.html.

“Adding Polynomials (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-arithmetic/x2ec2f6f830c9fb89:poly-add-sub/v/adding-and-subtracting-polynomials-1.

Multiplying Polynomials

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.
To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
The degree of a product of two polynomials equals the sum of the degrees of said polynomials.
The zeros of a product of two polynomial are the zeros of the two factors, combined.

References

“6.5 Multiplying Polynomials”. CK-12 Foundation. 2022. https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/6.5/primary/lesson/multiplying-polynomials-alg-ii/.

“Multiplying Polynomials (Steps and Solved Examples)”. 2022. BYJUS. https://byjus.com/maths/multiplying-polynomials/.

“Multiplying Polynomials”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/polynomials-multiplying.html.

“Multiplying Polynomials | Brilliant Math & Science Wiki”. 2022. brilliant.org. https://brilliant.org/wiki/polynomial-multiplication/.

“Multiplying Monomials (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-arithmetic/x2ec2f6f830c9fb89:mono-by-poly/v/multiply-monomials-intro.

“Multiplying Binomials By Polynomials: Area Model (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-arithmetic/x2ec2f6f830c9fb89:bi-by-poly/v/multiplying-polynomials-using-area-model.

⭐ “Polynomial Special Products: Difference Of Squares (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-arithmetic/x2ec2f6f830c9fb89:special-products/v/poly-diff-of-squares.

Dividing Polynomials

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials.

Long Division

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat.

Set up the division problem.
Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
Multiply the answer by the divisor and write it below the like terms of the dividend.
Subtract the bottom binomial from the terms above it.
Bring down the next term of the dividend.
Repeat steps 2–5 until reaching the last term of the dividend.
If the remainder is non-zero, express as a fraction using the divisor as the denominator.

For example, if we were to divide 2x³−3x²+4x+5 by x+2 using the long division algorithm, it would look like this:

The Division Algorithm

The Division Algorithm states that given a polynomial dividend f(x) and a non-zero polynomial divisor d(x) where the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that

f(x) = d(x)q(x)+r(x)

q(x) is the quotient and r(x) is the remainder. The remainder is either equal to zero or has degree strictly less than d(x).

If r(x)=0, then d(x) divides evenly into f(x). This means that both d(x) and q(x) are factors of f(x).

Synthetic Division

Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. One of the advantages of using this method over the traditional long method is that the synthetic division allows one to calculate without writing variables while performing the polynomial division, which also makes it an easier method in comparison to the long division.

Synthetic division of polynomials uses numbers for calculation and avoids the usage of variables. In the place of division, we multiply, and in the place of subtraction, we add.

Write the coefficients of the dividend and use the zero of the linear factor in the divisor’s place.
Bring the first coefficient down and multiply it with the divisor.
Write the product below the 2nd coefficient and add the column.
Repeat until the last coefficient. The last number is taken as the remainder.
Take the coefficients and write the quotient.
Note that the resultant polynomial is of one order less than the dividend polynomial.

Consider this division: (x³ – 2x³ – 8x – 35)/(x – 5). The polynomial is of order 3. The divisor is a linear factor. Let’s use synthetic division to find the quotient. Thus, the quotient is one order less than the given polynomial. It is x² + 3x + 7 and the remainder is 0. (x³ – 2x³ – 8x – 35)/(x – 5) = x² + 3x + 7.

*How to do Synthetic Division – CUEMATH*

References

“2.4 Division Algorithm for Polynomials – Statement, Interactive and Examples”. CK-12 Foundation. https://flexbooks.ck12.org/cbook/ck-12-cbse-math-class-10/section/2.4/primary/lesson/division-algorithm-for-polynomials/.

“6.8 Factoring Polynomials in Quadratic Form”. 2022.CK-12 Foundation. https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/6.10/primary/lesson/synthetic-division-of-polynomials-alg-ii/.

“6.9 Long Division of Polynomials”. CK-12 Foundation. 2022. https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/6.9/primary/lesson/long-division-of-polynomials-alg-ii/.

“Dividing Polynomials | College Algebra”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-dividing-polynomials/.

“Division Algorithm – Formula, For Polynomials, Examples”. 2022. CUEMATH. https://www.cuemath.com/algebra/division-algorithm-for-polynomials/.

“Division Of Polynomial By Linear Factor | Solved Examples | Algebra- CUEMATH”. 2022. CUEMATH. https://www.cuemath.com/algebra/division-of-polynomial-by-linear-factor/.

“Long Division Polynomial – Definition, Method, Long Division With Monomials, Binomials”. 2022. CUEMATH. https://www.cuemath.com/algebra/long-division-of-polynomials/.

“Polynomial Expressions, Equations, & Functions | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra-home/alg-polynomials.

“Polynomials – Long Division”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/polynomials-division-long.html.

“Synthetic Division – Method, Steps, Examples, FAQs”. 2022. CUEMATH. https://www.cuemath.com/algebra/synthetic-division-of-polynomial/.

Definitions

Polynomials are expressions involving x raised to a whole number power (exponent).

Degree of a polynomial: The highest power (exponent) of x.

Relative maximum: The point(s) on the graph which have maximum y values or second coordinates “relative” to the points close to them on the graph.

Relative minimum: The point(s) on the graph which have minimum y values or second coordinates “relative” to the points close to them on the graph.

Absolute maximum: The point on the graph which has the largest value of y (the second coordinate of the point).

Absolute minimum: The point on the graph which has the smallest value of y (the second coordinate of the point).

Courses

“Unit: Polynomial expressions, equations, & functions”. 2022. Khan Academy. https://www.khanacademy.org/math/algebra-home/alg-polynomials.

Videos

What is a Polynomial? | ExamSolutions

Here you will learn about polynomials and the degree of a polynomial

The featured image on this page is from the YouTube video “What is a Polynomial?” 2012. ExamSolutions. https://youtu.be/lYy8Nh9AdWk.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.