Polynomials: adding, subtracting and multiplication

Polynomials: adding, subtracting and multiplication

Like terms

• Chapter 2 we discussed the combining like terms.

• Like terms mean the variable and exponent must be the same in order to combine.

• Coefficients were the numerical value found in the front of the term.

• Ex: 4x²; 4 is the coefficient, x is the variable and 2 is the exponent.

• Like terms have may have differing coefficients but not variable exponents.

Simplifying like terms

• Terms can be added or subtracted from each other if they are exactly alike.

• Ex. 9mp + 7 nq - 5mp

• Only the 9mp and 5mp can be combined

• 4mp + 7nq would be the answer.

Fill in the Blank

Simplify by Combining Like Terms:

10a - 5 -12a + 14

Fill in the Blank

Simplify by Combining Like Terms:

-2 - 4x + 6 - 4

Fill in the Blank

Simplify by Combining Like Terms:

-8b + 3b + b

Fill in the Blank

Simplify by Combining Like Terms:

3x + 6x

Fill in the Blank

Simplify by Combining Like Terms:

3 + 6 + 3a

Multiple Choice

Which is not a Like Term?

• 2x

• -13x

• -13

• -2x

Polynomials

• Polynomials are classified as:

• Monomial: one term 3m, -6x², 5xy

• Binomial: two terms x - 5; 3t + 10

• Trinomials: three terms x² + 3x + 3

• Always combine like terms before classifying the type of polynomial

• Polynomials are always written in descending order of exponent value.

Degree of term

• The degree of term is the sum of the exponents on the variables. Ex. 3x⁴ the term is in the 4th degree

• Degree of polynomial is the greatest degree of any nonzero term of the polynomial.

• Ex: 3m⁵ + 5m² - 2m + 1; The degree of polynomial is 5 because that is the greatest exponent value.

• Make sure you know the difference between the two terms.

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Multiple Choice

What is the degree?
 - 3x³ - x² - 10x + 12

• 1
• 2
• 3
• 4

Multiple Choice

Classify the polynomial:

3x² – 8x + 1

• quadratic trinomial

• cubic trinomial

• quadratic binomial

• cubic binomial

Multiple Choice

Classify the following polynomial

• quadratic trinomial
• cubic binomial

Multiple Choice

Classify by its degree

• Degree 3

• Degree 6

• Degree 7

• Degree 5

Multiple Choice

Classify by its degree

• Degree 1

• Degree 2

• Degree 3

• Degree 4

Multiple Choice

Classify by its degree

• Degree 2

• Degree 3

• Degree 4

• Degree 5

Multiple Choice

Classify by its degree

• Degree 1

• Degree 2

• Degree 4

• Degree 5

Multiple Choice

Classify by the number of terms

• Monomial

• Binomial

• Trinomial

• Polynomial

Multiple Choice

Classify by number of terms:
7x³ – 8x² + 9

• Monomial
• Binomial
• Trinomial
• 4-Term Polynomial

Multiple Choice

Classify by number of terms:

2x – 9

• Monomial

• Binomial

• Trinomial

• 4-Term Polynomial

Evaluating polynomials

• substitute the given value and solve the expression.

• Ex: 2x² + 8x -6 for x = -4

• 2(-4)² + 8(-4) - 6

• 32 - 32 - 6

• Answer -6

Multiple Choice

Given

f(x)= 2x² + 5x - 17, find f(-1).

• f(-1) = -20

• f(-1) = -24

• f(-1) = -17

• f(-1) = -10

Multiple Choice

If f(x) = x²+ 3, find f(-2)

• -1

• 1

• 7

• -7

Adding polynomials

• This can be done horizontally or vertically.

• Combine like terms to obtain the sum.

• Add: 6x³ - 4x² + 3 and - 2x³ + 7x² -5

• Horizontal will allow you to see the same terms clearly.

• 6x³ - 4x² + 3

• -2x³ + 7x² -5

• 4x³ + 3x² -2 each term has the sign in front as it's integer sign.

Multiple Choice

 $\left(2-n^2+3n\right)+\left(4n^2+6-8n\right)$  

•  $5n^2+11n+8$  

•  $3n^2-5n+8$  

•  $3n^2+5n-4$  

•  $5n^2+3n-11$  

Multiple Choice

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

•  $3a^2+8a$  

•  $a^2-8a$  

•  $3a^2-2a$  

•  $3a^2-8a$  

Multiple Choice

 $\left(2x^2-5x+6\right)+\left(3x^2+x-7\right)$  

•  $5x^2-4x-1$  

•  $5x^2-6x+13$  

•  $x^2-4x+13$  

•  $5x^2+4x-1$  

Multiple Select

When combine like terms what arithmetic do we perform on the exponents? (Click all that apply)

• Addition

• Subtraction

• Multiplication

• Division

• Nothing, we leave the exponents as they are

Subtracting polynomials

• Most important fact is that you use distribution to change the second polynomial by multiplying by the negative value.

• Ex. (3x - 8) - (5x - 9)

• Distribute: 3x - 8 + (-5x + 9) we did this in chapter 1 when we talked about distributing the negative term. Now add the polynomials.

• 3x - 8

• +-5x -9

• -2x - 17

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Multiple Choice

(5x⁴ - 4x³ - 2x² + x - 19) - (x⁴ + 5x³ + 8x² + x + 5)

• 4x⁴ - 9x³ - 10x² - 24
• 4x⁴ - 9x³ - 10x² - 2x - 24
• 4x⁴ - 9x³ - 10x² + 2x - 24
• -4x⁴ - 9x³ - 10x² - 24

Multiple Choice

(3x + 4y - 3z) - (2x - 6y + 7z)

• x + 10y - 10z
• -x +10y - 10z
• x -10y - 10z
• -x -10y - 10z

adding and subtracting polynomials in same expression

• Always do the combining with two polynomials at a time.

• Remember to distribute the negative sign to change the subtraction to addition.

• This is just like performing single variable polynomial work.

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Multiplying polynomials

• Use distribution to multiply polynomials. Use the exponent rules when solving.

• Ex. 4x( 3x² + 2)

• 4x(3x²) + 4x(2)

• 12x³ + 8x

• Always give final answer in descending exponent order.

• Ex: (x - 4)(x² + 3x -5)

• Distribute each term:

• x(x²) +x(3x) -x(5) -4(x²) -4(3x) -4(-5)

• x³ + 3x² -5x -4x² -12x +20

• combine like terms to complete.

• x³ -x² -17x +20

• Make sure final answer is in descending order. It is!

Multiple Choice

 $\left(5v+4\right)\left(3v-6\right)$  

•  $15v^2+24v-18$  

•  $15v^2-18v-24$  

•  $15v^2-34v+24$  

•  $15v^2-24v+18$  

Multiple Choice

2x (–2x – 3)

• –4x – 3

• x² – 3

• –4x² – 6x

• –4x – 6

Multiple Choice

Simplify:
5xy(6x-y)

• 11x² - y²
• 30x²y - y
• 30x²y - 5xy²
• 11x²y + 5x²y²

Multiple Choice

Select the expanded form of the expression:

(x - 2)(x + 3)

• (x + 1)

• (x - 6)

• (x + 6)

• (x² + x + 1)

• (x² + x - 6)

Special products

• squaring binomials

• (x+y)² = x² 2xy + y² is the formula

• Let's try it.

• (t +6)² = t² + 12t + 36

• (x -y)² = x² -2xy + y²

• (2m - p)² = 4m² -4mp + p²

Other special products

• (x + y)(x-y) = x² -y²

• These are conjugates of the same terms.

• Read page 399 in textbook for the greater powers of binomials.

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Multiple Choice

Challenge: (3x + 2)³ = ___________________. Hint: 2 seperate box problems.

• 27x³ + 18x² + 36x + 8

• 27x³ + 54x² + 36x + 6

• 27x³ + 54x² + 36x + 8

• 27x³ + 54x² + 24x + 8

Multiple Choice

(x + 5)²

• x² + 10x + 25

• x² + 5x + 25

• x² + 6x - 10

• x² + 25

Multiple Choice

(7r-6)(7r+6)

• 49r²+36

• 49r²-36

• 49r²+84r+36

• 49r²+42r+36

Multiple Choice

(2u-4v)²

• 4u²-8uv+16v²
• 4u²-16uv+16v²
• 4u²-16v²
• 4u²+8uv+16v²

Multiple Choice

(x+4)²

• x²+16
• x²-16
• x²+8x+16
• x²+8x-16