1. Polynomials and Special Products
Section P.3
Polynomials and Special Products

2. Definition of a Polynomial in x
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree
Polynomial Function

3. Definition of a Polynomial in x
Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer.
A polynomial in x is an expression of the form
Where an ≠ 0.
The polynomial is of degree n.
The leading coefficient is an
The constant term is a0.

4. Definition of a Polynomial in x
Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer.
A polynomial in x is an expression of the form
Where an ≠ 0.
Is this a polynomial?

5. Definition of a Polynomial in x
Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer.
A polynomial in x is an expression of the form
Where an ≠ 0.
Is this a polynomial?

6. Names of Polynomials x2 – 16 x + 8 x4 + x3 – x – 15 7 x2 – 6x + 5
2nd degree, binomial
1st degree, binomial
4th degree, polynomial
Constant, monomial
2nd degree, trinomial
7th degree, binomial
x2 – 16
x + 8
x4 + x3 – x – 15 7
x2 – 6x + 5
x4y3 + 7

7. FOIL F O I L

8. Square a Binomial
Now here is a shortcut worth knowing.

9. Square a Binomial a2 Square the first term 2ab Double the product
These are directions for a shortcut in binomial multiplication.
a2 Square the first term
2ab Double the product
of the terms
b2 Square the last term

10. Square a Binomial Negative Version
These are directions for a shortcut in binomial multiplication.
a2 Square the first term.
2ab Double the product
of the terms . Keep the sign
b2 Square the last term
Always positive.

11. Binomial Squared Examples
A) (t + u)2
B) (2m - p)2
C) (4p + 3q)2
D) (5r - 6s)2
E) (3k - 1/2)2
t2 + 2tu + u2
4m2 - 4mp + p2
16p2 + 24pq + 9q2
25r2- 60rs + 36s2

12. The Product of the Sum and Difference of Two Terms
First start with two terms.
Now write two binomials.
The other
a difference
One a sum

13. Binomial Sum and Difference Rule
When multiplying two binomials that differ only in the sign between their terms, subtract the square of the last term from the square of the first term.
Difference of Squares
Conjugate Factors =

14. Sum and Difference Examples
A) (6a + 3)(6a - 3)
B) (10m + 7)(10m - 7)
C) (7p + 2q)(7p - 2q)
D) (3r - 1/2)(3r + 1/2)
= 36a2 - 9
= 100m2 - 49
= 49p2 - 4q2

15. Cube a binomial

16. Apply the formula

17. Homework Page 30 1 – 6 9 – 42 multiples of 3
53, 59, 63, 71, 73, 86, 96, 97