Do Now:Evaluate each expression for x = -2.

Aim: How do we work with polynomials?

1) -x + 1

2) x2 - 5 3) -(x – 6)

Simplify each expression.

4) (x + 5) + (2x + 3)

5) (x + 9) – (4x + 6)

6) (-x2 – 2) – (x2 – 2)

Warm Up

Warm Up - Answers

What is the degree of the monomial? 245 bx

The degree of a monomial is the sum of the exponents of the variables in the monomial.

The exponents of each variable are 4 and 2. 4+2 = 6.

The degree of the monomial is 6.

The monomial can be referred to as a sixth degree monomial.

A polynomial is a monomial or the sum of monomials

24x 83 3 x 1425 2 xx Each monomial in a polynomial is a term of the

polynomial.

The number factor of a term is called the coefficient.

The coefficient of the first term in a polynomial is the lead coefficient.

A polynomial with two terms is called a binomial.

A polynomial with three terms is called a trinomial.

14 x

83 3 x

1425 2 xx

The degree of a polynomial in one variable is the largest exponent of that variable.

2 A constant has no variable. It is a 0 degree polynomial.

This is a 1st degree polynomial. 1st degree polynomials are linear.

This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.

This is a 3rd degree polynomial. 3rd degree polynomials are cubic.

To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term.

The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive.

745 24 xxx

x544x 2x 7

Write the polynomials in standard form.

243 5572 xxxx

32x4x 7x525x

)7552(1 234 xxxx

32x4x 7x525x

Remember: The lead coefficient should be positive in standard

form.

To do this, multiply the polynomial by –1 using the distributive

property.

The degree of a Monomial

Is the sum of the exponents of the variables of the monomial.

Monomial Degree

x

1x y 2

9 0

The degree of a Monomial

Is the sum of the exponents of the variables of the monomial.

x3 3x3 y2 5

Monomial Degree

3x3 y2 5

32x3 y2 5

The degree of a Polynomial

Is the highest degree of any of its terms after the poly has been simplified.

Polynomial Degree

3x2 + 5x + 7

2

Descending order of Polynomials

From the highest degree to the lowest degree of the terms.

3x2 + 5x + 7 3x3 + 5x2 - 2x + 7

3. Find the perimeter of the triangle.

6a 5 3a

3a 2

P = (6a - 5) + (3a + 2) + 3a

P = 12a - 3

4. 4x3 3x 10x 2 8x2 2x x 3 x4

Combine like terms and put terms in descending order

4x3 3x 10x 2 8x2 2x x3 x 4

Simplify

x4 3x 3 2x2 5x

5. 2x2 3xy 5y2 4x 2 3xy 2y2 2x2 3xy 5y2 4x2 3xy 2y2

2x2 7y2

Simplify6. 9a2 3a 7b3

9a2 3a 7b3 9a2

27a3 63a2b3

7. x 8 x 12 x2 12x 8x 96x2 20x 96

8. x 3 2Simplify

x 3 x 3 x2 3x 3x 9

x2 6x 9

*Notice that (a+b) 2 = a2 +2ab +b2

Simplify9. 4c 5 2c 3

8c2 12c 10c 158c2 22c 15

10. 3y2 2 2y 1 6y3 3y2 4y 2

Simplify: (x + y) (x2 – xy + y2)

Simplify: (x – y) (x2 + xy + y2)

= x3 – x2y + xy2 + x2y – xy2 + y3

= x3 + x2y + xy2 – x2y – xy2 + y3

))(()( 223 yxyxyxyx

))(()( 223 yxyxyxyx

Note:

Simplify

2x3 24x2 10x 2 120x 4x 48

11. 2x2 10x 4 x 12

2x3 14x2 124x 48