college algebra

Module 2:  Polynomials

Polynomials

A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.

A. 3x2+2x+6 is a polynomial in one variable: X

B. x2y−6x−4 is a polynomial in two variables: x and y

C. 12x+1 is not a polynomial because the variable is in the denominator

Terms The parts that are added or subtracted in a mathematical expression are called terms. The degree of a term with whole-number exponents is the sum of the exponents of the variables, if there are variables. Nonzero constants have degree 0, and the term 0 has no degree.

A.  In the polynomial: 2x5+3x4+6x+1, the degree of the polynomial is 5 (from 2x5).

B.  In the polynomial: 8x2y4−6xy3+3xy2z4, the degree of the polynomial is 7 (from 3xy2z4).

Addition and Subtraction of Polynomials To add or subtract polynomials, remove parentheses if any are present. Then combine like terms. Example: Simplify (4𝑥2 − 6𝑥 + 8) + (2𝑥2 + 5𝑥 − 1)

1. remove parentheses: 4𝑥2 − 6𝑥 + 8 + 2𝑥2 + 5𝑥 − 1

2. rearrange terms: 4𝑥2 + 2𝑥2 − 6𝑥 + 5𝑥 + 8 − 1

3. combine like terms: 6𝑥2 − 𝑥 + 7

Multiplication of Polynomials

In mathematics, the asterisk symbol * refers to multiplication. For example, consider the following expression: 7 * 6 or 2*5(x2+3)

To multiply polynomials, you must remember that each term of one polynomial must be multiplied by each term of the other polynomial.

Also remember the product rule for exponents: 𝑎𝑚 * 𝑎𝑛 = 𝑎+𝑛 

Examples:

1. Multiply a Monomial by a Monomial: (2𝑥2)(5𝑥3) Remove parentheses and rearrange terms: 2.5*𝑥2*𝑥3 Product rule: 10𝑥2+3 = 10𝑥5

2. Multiply a Monomial by a Polynomial: (2𝑥)(3𝑥2 − 4𝑥) Multiply the monomial for each term of the polynomial: 2𝑥*3𝑥2 − 2𝑥*4𝑥 = 6𝑥3 − 8𝑥2

3. Multiply a Binomial by a Binomial: (3x + 2) (x – 5) Apply the distributive property: (a+b)(c+d) = (a+b)c + (a+b)d = ac + bc +ad +bd 3x*x + 2*x – 3x*5 – 2.5 = 3𝑥2 + 2𝑥 − 15𝑥 − 10 = 3𝑥2 − 13𝑥 − 10

Note: When multiplying two polynomials, every term of the first polynomial must be multiplied by every term of the second polynomial.

References

Angel, A., & Rundle, D.(2018).  Intermediate algebra for college students (10th edition). Pearson.

Module 2:  Polynomials

Polynomials

A polynomial is a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator.

A. 3x2+2x+6 is a polynomial in one variable: X

B. x2y−6x−4 is a polynomial in two variables: x and y

C. 12x+1 is not a polynomial because the variable is in the denominator

Terms The parts that are added or subtracted in a mathematical expression are called terms. The degree of a term with whole-number exponents is the sum of the exponents of the variables, if there are variables. Nonzero constants have degree 0, and the term 0 has no degree.

A.  In the polynomial: 2x5+3x4+6x+1, the degree of the polynomial is 5 (from 2x5).

B.  In the polynomial: 8x2y4−6xy3+3xy2z4, the degree of the polynomial is 7 (from 3xy2z4).

Addition and Subtraction of Polynomials To add or subtract polynomials, remove parentheses if any are present. Then combine like terms. Example: Simplify (4𝑥2 − 6𝑥 + 8) + (2𝑥2 + 5𝑥 − 1)

1. remove parentheses: 4𝑥2 − 6𝑥 + 8 + 2𝑥2 + 5𝑥 − 1

2. rearrange terms: 4𝑥2 + 2𝑥2 − 6𝑥 + 5𝑥 + 8 − 1

3. combine like terms: 6𝑥2 − 𝑥 + 7

Multiplication of Polynomials

In mathematics, the asterisk symbol * refers to multiplication. For example, consider the following expression: 7 * 6 or 2*5(x2+3)

To multiply polynomials, you must remember that each term of one polynomial must be multiplied by each term of the other polynomial.

Also remember the product rule for exponents: 𝑎𝑚 * 𝑎𝑛 = 𝑎+𝑛 

Examples:

1. Multiply a Monomial by a Monomial: (2𝑥2)(5𝑥3) Remove parentheses and rearrange terms: 2.5*𝑥2*𝑥3 Product rule: 10𝑥2+3 = 10𝑥5

2. Multiply a Monomial by a Polynomial: (2𝑥)(3𝑥2 − 4𝑥) Multiply the monomial for each term of the polynomial: 2𝑥*3𝑥2 − 2𝑥*4𝑥 = 6𝑥3 − 8𝑥2

3. Multiply a Binomial by a Binomial: (3x + 2) (x – 5) Apply the distributive property: (a+b)(c+d) = (a+b)c + (a+b)d = ac + bc +ad +bd 3x*x + 2*x – 3x*5 – 2.5 = 3𝑥2 + 2𝑥 − 15𝑥 − 10 = 3𝑥2 − 13𝑥 − 10

Note: When multiplying two polynomials, every term of the first polynomial must be multiplied by every term of the second polynomial.

References

Angel, A., & Rundle, D.(2018).  Intermediate algebra for college students (10th edition). Pearson.