Polynomials

Honors Math – Grade 8

Explore Find the area of the figure.

The total area is the area of the rectangle plus the area of the triangle.

5x

8x

x2

bhlwA21)(

Rectangle Formula Triangle Formula

2821)85( xxxxA

32 440 xxA This represents the total area of the figure.

Poly-what?

Polynomial

a monomial or sum of monomials

Examples:2x2 + 3x – 1 15a2b – c2

Non-example:

yx1

Made up of monomials

called terms

Terms are combined by

addition.

Polynomials with two terms are binomials and

those with three terms are trinomials.

Identify PolynomialsState whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial or trinomial.

Expression Polynomial? Identification?

yzx 32 23 58 nn

8954 2 aaa

YES. 2x – 3yz = 2x+(-3yz), the sum of two monomials.

NO. Not a monomial

22 55n

n

YES. -8 is a real number and all are monomials.

YES. The expression simplifies to 4a2 + 6a + 9, so it is the sum of three monomials.

Binomial

None

Monomial

Trinomial

Write a PolynomialGEOMETRY Write a polynomial to represent the area of the shaded region.

c

b

The area of the shaded region is the area of the rectangle minus the area of the circle.

)()( 2rlwA The length of the rectangle is b. The width of the rectangle is 2c.

)()2( 2rcbA

The radius of the circle is c.

)()2( 2ccbA 22 cbcA

The polynomial representing the area of the shaded region is: 22 cbcA

The degree of a monomial is the sum of the exponents of all its variables.

Monomial Degree

48ya3

41

322 zxy 1+2+3=67 0

jhg 5214 2+5+1=8

Polynomial Terms Degree of each Degree of poly

25mn 25mn 3 3534 222 xyx 5,3,4 222 xyx 4, 2, 0 416223 2 baaba 16,2,2,3 2baaba 1,2,3,0 3429 xxy 4,2,9 xxy 2,1,0 2

The degree of a polynomial is the greatest degree of any term in the polynomial.

256 abcrs 25,6 abcrs 2,4 4

Arrange Polynomials in Ascending OrderArrange the terms of each polynomial so that the powers of x are in ascending order.

The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending (decreasing) order.

1127 42 xx042 1127 xxx 42 2711 xx

Compare powers of x. 0 < 2 < 4

yxxyxy 2323 352 yxxyxyx 232031 352

Compare powers of x. 0 < 1 < 2 < 3

3232 532 xyxxyy

x0 = 1

x0 = 1 and x = x1

Arrange Polynomials in Descending OrderArrange the terms of each polynomial so that the powers of x are in ascending order.

32 2856 xxx

Compare powers of x. 3 > 2 > 1 > 0

xaaxaxa 25423 943 Compare powers of x. 5 > 2 > 1 > 0

x0 = 1

x0 = 1 and x = x13102 2856 xxxx

5862 23 xxx

1250423 943 xaaxxaxa 42235 934 axaxaax