Polynomial Review

Polynomial Flip Chart

ALGEBRA II 2020-2021

Polynomials

monomial: one term

binomial: two terms

trinomial: three terms

polynomial: more than three terms

EXAMPLES:

•   $7y\ -\ 3x\ +\ 4$  

•       There are 3 terms here, so this is called a TRINOMIAL

•  $10x^3yz^2$  

• •    There is only 1 term here, so this is called a MONOMIAL

• 
  

   $\frac{5}{2y^2}+7y$  

•        There are 2 terms here, so this is called a BINOMIAL

Multiple Choice

YOU TRY! Name the expression based on the number of terms it has.

2a + 3b² - 4c

• monomial

• binomial

• trinomial

Multiple Choice

YOU TRY!  Name the expression based on the number of terms it has.

 $4y\ -\frac{2}{6y^2}$  

• monomial

• binomial

• trinomial

Multiple Choice

YOU TRY! Name the expression based on the number of terms it has.

6rs³t⁶

• monomial

• binomial

• trinomial

DEGREE of a polynomial

• the highest degree of a monomial

• found by calculating the sum of the exponents of the variables that appear in it

EXAMPLES:

•  $5x^2$  

•       The variable has an exponent of 2, so the degree of the monomial is 2.

•  $4a^4b^3c$  

•       The variables a, b, & c have exponents of 4, 3, & 1 respectively. The sum of 4+3+1=8, so the degree of the monomial is 8.

•  $-3$  

•       There is NO variable in this monomial. The degree would be 0.

EXAMPLES:

•  $8x^2\ -\ 2x\ +\ 7$  

•      The trinomial has three terms, with exponents of 2, 1, and 0, respectively.  The highest exponent is 2, so the degree here is 2.

•  $y^7\ +\ 6y^4\ +\ 3x^4m^4$  

•       The polynomial has four terms, with degrees of 7, 4, and 8, respectively.  The highest one is 8, so the degree of this polynomial is 8.

Multiple Choice

YOU TRY! What is the degree of this monomial?

3p²

• 1

• 2

• 3

• 4

Multiple Choice

YOU TRY! What is the degree of this monomial?

7m³np⁴

• 3

• 4

• 7

• 8

Multiple Choice

YOU TRY! What is the degree of this monomial?

17

• 0

• 1

• 2

• 3

Multiple Choice

YOU TRY! What is the degree of this trinomial?

-9x² - 5x + 2

• 0

• 1

• 2

• 3

Multiple Choice

YOU TRY!   What is the degree of this polynomial?

 $4f^3\ -\ 7f^2g^2\ +\ 3g^2$  

• 1

• 2

• 3

• 4

Ascending vs Descending Order

Ascending Order: Listing the terms from SMALLEST degree to LARGEST degree

Descending Order: Listing the terms from LARGEST degree to SMALLEST degree

EXAMPLE 1: Write the following polynomial in ASCENDING ORDER.

 $8x\ -\ 3x^2\ +\ x^4\ -\ 4$  

• The terms have degree of 1, 2, 4, and 0, respectively.  To write this from SMALLEST to LARGEST, you would get:

•  $-4\ +\ 8x\ -\ 3x^2\ +\ x^4$  

EXAMPLE 2: Write the following polynomial IN TERMS OF X in DESCENDING ORDER.

 $12x^2y^3\ -\ 6x^3y^2\ +\ 3y\ -2x$  

• Since this says "in terms of x", ONLY focus on the exponents on the x-values.  List these exponents from LARGEST to SMALLEST degree.

• The degree on the "x's" are 2, 3, 0, and 1, respectively.

•  $-6x^3y^2\ +\ 12x^2y^3\ -\ 2x\ +\ 3y$  

Multiple Choice

YOU TRY!  Write this polynomial in ASCENDING ORDER.

 $3d\ +\ 4d^4\ -\ 5d^2\ +7$  

•  $4d^4\ -\ 5d^2\ +\ 3d\ +\ 7$  

•  $7\ +\ 3d\ \ -\ 5d^2\ +\ 4d^4$  

•  $-5d^2\ +\ 3d\ +\ 4d^4\ +\ 7$  

Multiple Choice

YOU TRY!  

Which polynomial is written in DESCENDING order in terms of y?

• $8xy^2\ +\ 3x^2y\ -\ 2x^3$

• $6x^3y\ -\ 7x^2y^3\ +\ 5xy^4$

ADDING & SUBTRACTING POLYNOMIALS

Simply Combine Like Terms!!!

• Use the commutative property to group like terms

• Add or subtract coefficients

Example 1: (9y - 7x + 15a) + (-3y + 8x - 8a)

• Group like terms and add or subtract coefficients

• 9y + (-3y) = 6y

• -7x + 8x = 1x = x

• 15a + (-8a) = 7a

• FINAL ANSWER: 6y + x + 7a

Example 2: (3a² + 3ab - b²) + (4ab + 6b²)

• Group like terms and add or subtract coefficients

• 3ab + 4ab = 7ab

• -b² + 6b² = 5b²

• (The 3a² term has nothing to combine with so just bring it down in the final answer)

• FINAL ANSWER : 3a² + 7ab + 5b²

EXAMPLE 3:

              $4x^2\ -\ 2xy\ +\ 3y^2$  

 $\left(-\right)\ \left(-3x^2\ -\ xy\ +\ 2y^2\right)$  

• To SUBTRACT these terms, make sure you DISTRIBUTE the "minus" sign to ALL terms in the second polynomial.

•        $4x^2\ -\ \left(-3x^2\right)\ =\ 7x^2$  

•        $-2xy\ -\ \left(-1xy\right)\ =\ -1xy$  

•        $3y^2\ -2y^2\ \ =\ \ 1y^2$  

• FINAL ANSWER    $7x^2\ -\ xy\ +\ y^2$  

Multiple Choice

YOU TRY!  Add these polynomials

 $\left(3y^3\ -\ 6y^2\ +\ 5y\right)\ +\ \left(2y^4\ -\ 4y^2\right)$  

•  $2y^{4^{ }}+\ 3y^3\ -\ 10y^2\ +\ 5y$  

•  $3y^3\ -\ 8y^{4^{\ }\ }+\ 5y\$  

•  $3y^3\ -\ 10y^2\ +\ 7y^5$  

Multiple Choice

YOU TRY!  Subtract these polynomials.

 $\left(w^2\ -\ 7w\ +\ 4\right)\ -\ \left(w^2\ -\ 2w\ -\ 8\right)$  

•  $-5w\ -\ 4$  

•  $-5w\ +\ 12$  

•  $-9w\ -\ 4$  

•  $2w^2\ -\ 9w\ -\ 4$  

Multiplying Monomials

"keep the base, add the exponents"

EXAMPLE:

 $x^2\ \cdot\ x^4$  

• Use the Product Rule:  "keep the base and ADD the exponents"

•     $x^{\ \left(2+4\right)}\ =\ x^6$  

• FINAL ANSWER:      $x^6$  

Multiple Choice

YOU TRY!

Multiply these monomials.

 $x^3\ \cdot\ x^7$  

•  $x^{10}$  

•  $x^{21}$  

Multipying Monomials

"keep the base, multiply the exponents"

EXAMPLE:   $\left(x^2\right)^3$  

• Use the Power Rule:  "keep the base and MULTIPLY the exponents"

•  $x^{\left(2\cdot3\right)}\ =\ x^6$  

• FINAL ANSWER:     $x^6$  

Multiple Choice

YOU TRY!
Multiply.

 $\left(x^7\right)^2$  

•  $x^9$  

•  $x^{14}$  

Multiplying Monomials

"distribute the outside exponent to EACH term inside the parenthesis"

Use the Power Rule when necessary

EXAMPLE:   $\left(rs\right)^4$  

• Use the Power of a Product Rule:  "Distribute the outside exponent to EACH term inside the parenthesis"

•  $r^4\ \cdot\ s^4$  

• FINAL ANSWER:      $r^4s^4$  

Multiple Choice

YOU TRY!  Simplify.  

 $\left(2x^3y\right)^2$  

•  $4x^5y^2$  

•  $4x^6y^2$  

•  $2x^5y^2$  

•  $4x^6y^2$  

"any term raised to the '1st' power is ALWAYS equal to that term"

Multiplying Monomials

"ANY term raised to the '0' power is ALWAYS equal to 1"

EXCEPTION: 

 $n\ \ne\ 0$  

Multiple Choice

 YOU TRY! Solve.

 $4x^2y^0\ \cdot\ 3x^0y^3z$  

•  $12x^2y^3z$  

•  $7x^2y^3$  

•  $12x^3y^4z$  

AKA "move it and lose it"

"move" the term being raised to the negative exponent to the OPPOSITE side of the fraction bar and "lose" the negative sign on the exponent

EXAMPLE: Simplify using the negative exponent rule.

 $4\ ^{-3}$  

• Write any whole number as a fraction.   $\frac{4^{-3}}{1}$  

• Apply the "move it and lose it" rule

• Move the 4 to the bottom of the fraction and lose the negative on the exponent of -3

• FINAL ANSWER:    $\frac{1}{4^3}$  =   $\frac{1}{64}$  

Multiple Choice

YOU TRY! Simplify . Do not leave negative exponents.

 $\frac{2\ x^{-3\ }y^0}{3\ x^{-4}\ y}$  

•  $\frac{3x^4}{4y}$  

•  $\frac{2xy^7}{1}$  

•  $\frac{2x}{3y}$  

DIVIDING POLYNOMIALS

"keep the base, subtract the exponents"

EXAMPLE: Divide.

 $\frac{x^5}{x^2}$  

• Use the division rule for exponents:  "keep the base, and subtract the exponents"

•  $x^{\left(5-2\right)}\ =\ x^3$  

• FINAL ANSWER:    $x^3$  

Multiple Choice

YOU TRY! Simplify by Dividing

 $\frac{4x^7}{8x^2}$  

•  $\frac{x^5}{2}$  

•  $\frac{x^5}{-4}$  

•  $\frac{x^9}{2}$  

Multiply Polynomial by Monomial

• 1) Distribute

• 2) Multiply coefficients

• 3) Add powers of like bases

• 4) Combine like terms

EXAMPLE:

 $4x\left(2xy\ -\ 8x^2\right)$  

• Distribute the outside term into each term in parenthesis

•        $4x\ \cdot\ 2xy\ =\ 8x^2y$  

•           $4x\ \cdot\ -8x^2\ =\ -32x^3$  

• FINAL ANSWER:    $8x^2y\ -\ 32x^3$  

Multiple Choice

YOU TRY! Distribute.

 $-6xy^2\ \left(xy\ +\ 2x^2\right)$  

•  $-6xy^2\ -\ 8x^2y$  

•  $-6x^2y^3\ -\ 12x^3y^2$  

•  $-12x^4y^3$  

WORK

 $-6xy^2\ \left(xy\ +2x^2\right)$  

• Distribute the outside term into every term inside the parenthesis

•       $-6xy^2\ \cdot\ xy\ \ =\ \ -6x^2y^3$  

•       $-6xy^2\ \cdot\ 2x^2\ \ =\ \ -12x^3y^2$  

• FINAL ANSWER:    $-6x^2y^3\ -\ 12x^3y^2$   

Multiple Choice

YOU TRY! Distribute.

 $-4y^2\ \left(5y^4\ -\ 3y^2\ +\ 2\right)$  

•  $-20y^8\ +\ 12y^4\ -\ 8y^2$  

•  $-9y^6\ +\ 7y^4\ -\ 6y^2$  

•  $-20y^6\ +\ 12y^4\ -\ 8y^2$  

WORK

 $-4y^2\ \left(5y^{4\ }-\ 3y^2\ +\ 2\right)$  

• Distribute the outside term into every term inside the parenthesis

•        $-4y^2\ \cdot\ 5y^4\ \ =\ \ -20y^6$  

•        $-4y^2\ \ \cdot\ \ -3y^2\ \ =\ 12y^4$  

•          $-4y^2\ \cdot\ 2\ \ =\ \ -8y^2$  

• FINAL ANSWER:   $-20y^6\ +\ 12y^4\ -\ 8y^2$  

MULTIPLYING BINOMIALS

• 1) Double Distribution

• 2) FOIL (First, Outer, Inner, Last)

• 3) Box Method

DOUBLE DISTRIBUTION

 $\left(2x\ +\ 3\right)\left(5x\ +\ 8\right)$  

• Distribute 2x into (5x + 8)

•         $2x\ \cdot\ 5x\ =\ 10x^2\ \ \ \ and\ \ \ 2x\ \cdot\ 8\ =\ 16x$     

•  Distribute 3 into (5x + 8)  
  

•          $3\ \cdot\ 5x\ =\ 15x\ \ \ \ \ and\ \ \ \ \ 3\ \cdot\ 8\ =\ 24$   

•   Put all together:  $10x^2\ +\ 16x\ +\ 15x\ +\ 24\ \$  

• Combine like terms and simplify

• FINAL ANSWER:   $10x^2\ +\ 31x\ +\ 24$  

FOIL (First-Outer-Inner-Last)

 $\left(y\ +\ 3\right)\left(y\ +\ 7\right)$  

• F:     $y\ \cdot\ y\ =\ y^2$  

• O:    $y\ \cdot\ 7\ =\ 7y$  

• I:      $3\ \cdot\ y\ =\ 3y$  

• L:      $3\ \cdot\ 7\ =\ 21$  

•  $y^2\ +\ 7y\ +\ 3y\ +\ 21\$  

• Combine like terms and simplify

• FINAL ANSWER:   $y^2\ +\ 10y\ +\ 21$  

BOX METHOD

 $\left(5x\ +\ 2\right)\left(3x\ -\ 5\right)$  

• Multiply inside the boxes

•  $15x^2\ -\ 25x\ +\ 6x\ -\ 10$  

• Combine Like Terms and Simplify

• FINAL ANSWER:   $15x^2\ -\ 19x\ -\ 10$  

Multiple Choice

Multiply using whatever method you prefer.

 $(x-9)(x+4)$  

•  $x^2\ +\ 13x\ -\ 36$  

•  $x^2\ -\ 5x\ -\ 36$  

•  $2x\ -\ 5x\ -\ 5$  

•  $x^2\ -\ 5x\ +\ 36$  

Multiple Choice

Multiply using whatever method you prefer.

 $(2x-5y)(x+3y)$  

•  $2x^2\ +\ 11xy\ -\ 15y^2$  

•  $2x^2\ -\ 15y^2$  

•  $2x^2\ +\ xy\ -\ 15y^2$  

Divide Polynomial by Monomial

• Divide ALL expressions in the numerator by the denominator

Divide.   $\frac{6x^3y^3\ -\ 10x^5y}{2x^2y}$  

• Divide each term in the numerator by the monomial in the denominator.

•         $\frac{6x^3y^3}{2x^2y}\ \ =\ \ 3xy^2\ \ \ \ \ and\ \ \ \ \ \frac{-10x^5y}{2x^2y}\ =\ -5x^3$  

• FINAL ANSWER:   $3xy^2\ -\ 5x^3$     

Multiple Choice

YOU TRY! Divide.

 $\frac{4a^3bc^0\ +\ 8ab^3c}{16ab}$  

•  $\ \frac{3a^3b^3c}{4}$  

•  $\frac{a^2b^2c}{6}\$  

•  $\frac{a^2}{4}\ +\ \frac{b^2c}{2}$  

Multiplying Polynomials    

•  $\left(2x\ -\ 5\right)\left(x^2\ -\ 5x\ +\ 4\right)$      

•  Use an "extended" box method to solve.
  

•   Combine like terms and simplify

• FINAL ANSWER:    $2x^3\ -\ 15x^2\ +\ 33x\ -\ 20$   

Multiple Choice

YOU TRY!  Multiply.

 $\left(2p\ +\ 1\right)\left(p^2\ -\ 3p\ +\ 4\right)$  

•  $2p^3\ -\ 5p^2\ +\ 5p\ +\ 4$  

•  $2p^3\ +\ 7p^2\ +\ 11p\ +\ 4$  

Special Products

• 1. Multiplying a Binomial by Itself

• What happens when we square a binomial (in other words, multiply it by itself) .. ?

• (a+b)² = (a+b) (a+b) = ... ?

Special Products

• 2. Subtract Times Subtract

• What happens when we square a binomial with a minus inside?

• (a−b)² = (a−b) (a−b) = ... ?

Special Products

• 3. Add Times Subtract

• There is one more special case ... what about (a+b) times (a−b) ?

• (a+b) (a−b) = ... ?

• This is called the "difference of two squares" 

Multiple Choice

YOU TRY!  Simplify. $\left(x\ +\ 4\right)\ ^2$  

•  $x^2\ +\ 16x\ +\ 16$  

•  $x^2\ +\ 16$  

•  $x^2\ +\ 8x\ +\ 8$  

•  $x^2+\ 8x\ +\ 16$  

Multiple Choice

YOU TRY!  Simplify.     $\left(x\ -\ 5\right)\ ^2$  

•  $x^2\ -\ 25$  

•  $x^2\ +\ 25$  

•  $x^2\ -\ 10x\ +\ 25$  

•  $x^2\ -25x\ +\ 25$  

Multiple Choice

YOU TRY! Simplify.     $\left(x\ +\ 3\right)\ \left(x\ -\ 3\right)$  

•  $x^2\ -\ 9$  

•  $x^2\ -\ 6x\ -\ 9$  

•  $x^2\ -\ 6x\ +\ 9$  

•  $x^2\ +\ 9$  

Poll

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