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