Download Section 0.4 Polynomials

Section 0.4 Polynomials
A monomial is a number or the product of a number and one or more variables with wholenumber exponents. The number is called the coefficient of the variables. The degree of a monomial is the sum of the exponents of its variables. All nonzero constants have a degree of 0.
Ex: Some monomials:
• 3x where 3 is the coefficient and the degree is 1.
• 7ab2 where 7 is the coefficient and the degree is 3.
• −17a4 b2 c where 17 is the coefficient and the degree is 7.
• y 3 where 1 is the coefficient and the degree is 3.
• −82 where −82 is the coefficient and the degree is 0.
NOTE: The monomial 0 has no defined degree.
Polynomials
A monomial or sum of monomials is called a polynomial. Each monomial in the sum is a term
of the polynomial. A polynomial with two terms is a binomial and a polynomial with three terms
is a trinomial.
Monomials
6x3
−16ab
a3 b 2 c 4
Binomials
4x + 3y
9a3 − 5b2
−3a4 − b
Trinomials
x2 − 3x + 2
2y 2 + 5y − 17
2a2 b3 − 5ab − 13
The degree of a polynomial is the degree of the term in the polynomial with the highest degree. The polynomial 0, called the zero polynomial is the only polynomial with no defined degree.
Ex: Some polynomials and their degrees:
• 2x2 y 4 + 3xy − 9 is a trinomial of degree 6.
• 3ab + 4a2 b is a binomial of degree 3.
√
√
• 3x + 17y − 5z 4 − 3 10 is a polynomial of degree 4.
• −13y 1/2 − 4y + 3 is not a polynomial (all exponents must be whole numbers).
When two terms of a polynomial have the same variables with the same exponents, we say they
are like terms. We can combine like terms using the distributive property.
Ex: 2x2 y 3 + 3x2 y 3 = (2 + 3)x2 y 3 = 5x2 y 3 .
To combine like terms, add their coefficients and keep the same variables and exponents.
Adding and Subtracting Polynomials
To add and subtract polynomials, remove the parenthesis and combine like terms.
Recall: if a set of parenthesis has a + sign in front of it, drop the parenthesis. If a set of parenthesis
has a - sign in front of it, drop the parenthesis and change the sign of each term.
Ex: Add: (x3 − 3x2 ) + (5x3 − 8x).
We can simply remove the parenthesis because of the + sign. So we have:
x3 − 3x2 + 5x3 − 8x. Now, use the commutative property to rearrange terms:
x3 + 5x3 − 3x2 − 8x. Now combine like terms and simplify:
(1 + 5)x3 − 3x2 − 8x = 6x3 − 3x2 − 8x
Ex: Subtract: (3t7 − 7t3 + 3) − (7t7 − 3t3 + 7).
We cannot just remove the parenthesis because now we have a - sign. We must distribute the
imaginary negative 1 to every term in the second polynomial (i.e. change all signs):
3t7 − 7t3 + 3 − 7t7 + 3t3 − 7. Now, use the commutative property to rearrange terms:
3t7 − 7t7 − 7t3 + 3t3 + 3 − 7. Now combine like terms and simplify:
(3 − 7)t7 + (−7 + 3)t3 + (3 − 7) = −4t7 − 4t3 − 4
To add multiples of one polynomial to another, or to subtract multiples of one polynomial from
another, remove the parenthesis and add like terms.
Ex: Simplify: 5(x3 − 8x + 3) + 2(3x2 + 5x) − 7.
We must distribute the constants into the corresponding polynomial:
5x3 − 40x + 15 + 6x2 + 10x − 7. Now, use the commutative property to rearrange terms:
5x3 + 6x2 − 40x + 10x − 7. Now combine like terms and simplify:
5x3 + 6x2 + (−40 + 10)x − 7 = 5x3 + 6x2 − 30x − 7.
Multiplying Polynomials
• To multiply two monomials, multiply the coefficients and then multiply the variables.
• To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.
Ex: Find the product of 3xy 2 z and 6x2 yz 3 .
These are both monomials. First, multiply the coefficients. Then multiply the variables and
simplify:
(3xy 2 z)(6x2 yz 3 ) = 3 · 6 · x · x2 · y 2 · y · z · z 3 = 18 · x1+2 · y 2+1 · z 1+3 = 18x3 y 3 z 4 .
(We can do this because we can use the commutative property to rearrange terms.)
Ex: Find the product of 2x2 y and 3xy + y 2 − 5yz.
This will be the product of a monomial and a trinomial. We must multiply each term in the
trinomial by the monomial (we are using the distributive property):
(2x2 y)(3xy + y 2 − 5yz) = (2x2 y)(3xy) + (2x2 y)(y 2 ) − (2x2 y)(5yz)
Now, each term is just the product of two monomials, so we can proceed as in the previous
example: = 2 · 3 · x2 · x · y · y + 2 · x2 · y · y 2 − (2 · 5 · x2 · y · y · z) = 6 · x1+2 · y 1+1 + 2 · x2+0 ·
y 1+2 − (10 · x2+0 · y 1+1 · z 0+1 ) = 6x3 y 2 + 2x2 y 3 − 10x2 y 2 z.
• Special Product Formulas:
(x + y)2 = (x + y)(x + y) = x2 + 2xy + y 2
(x − y)2 = (x − y)(x − y) = x2 − 2xy + y 2
(x + y)(x − y) = x2 − y 2
NOTE: (x + y)2 6= x2 + y 2 and (x − y)2 6= x2 − y 2 . The products are trinomials, not binomials.
• We can multiply a binomial by a binomial in two different ways.
– We can multiply the first term in the first binomial by every term in the second binomial
and then multiply the second term in the first binomial by both terms in the second
binomial and then combine like terms.
– FOIL Method to multiply two binomials: FOIL stands for First terms, Outer terms,
Inner terms, Last terms. We multiply the corresponding terms, then add them.
Ex: Multiply (2x + 3)(4x − 5).
– First terms: 2x and 4x
– Outer terms 2x and −5
– Inner terms: 3 and 4x
– Last terms: 3 and −5
Now FOIL:
(2x)(4x) + (2x)(−5) + (3)(4x) + (3)(−5) = 8x2 − 10x + 12x − 15 = 8x2 + 2x − 15
Ex: Multiply (3x −
√
5)(x +
√
5).
√
√
√ √
√
√
√
FOIL: (3x)(x)+(3x)( 5)+(− 5)(x)+(− 5)( 5) = 3x2 +3 5x− 5x−5 = 3x2 +2 5x−5
• To multiply a polynomial with more than two terms by another polynomial, multiply each
term of one polynomial by each term of the other polynomial. Then combine like terms and
simplify.
Ex: Multiply (a + b)(a2 − ab + b2 ).
We multiply each term of the second polynomial by a and then by b and then add all the
terms:
(a+b)(a2 −ab+b2 ) = a(a2 )+a(−ab)+a(b2 )+ba2 +b(−ab)+b(b2 ) = a3 −a2 b+ab2 +a2 b−ab2 +b3 .
Now combine like terms:
= a3 + b 3
• We can multiply polynomials together if the exponent n is a whole number.
Ex: (xn + 3)(2xn − 1) = (xn )(2xn ) + (xn )(−1) + (3)(2xn ) + (3)(−1) = 2x2n − xn + 6xn − 3 =
2x2n + 5xn − 3
• We can multiply expressions that are not polynomials by using previous methods:
Ex: Multiply (x−3 + 2y)(x3 − y −1 ).
Note that these expressions are not polynomials because the exponents are not all whole numbers. However, we can still proceed as before:
(x−3 + 2y)(x3 − y −1 ) = (x−3 )(x3 ) + (x−3 )(−y −1 ) + (2y)(x3 ) + (2y)(−y −1 ) = x0 − x−3 y −1 +
2x3 y − 2y 0 . Recall that x0 = 1 and y 0 = 1:
= 1 − x−3 y −1 + 2x3 y − 2 = −x−3 y −1 + 2x3 y − 1.
Conjugate Binomials
Conjugate binomials are binomials that are the same except for the sign between their terms.
The conjugate of a + b is a − b, and the conjugate of a − b is a + b.
If we have a fraction whose denominator is a binomial with square roots, we can use conjugate
binomials to rationalize the denominator. Likewise, we can use conjugate binomials to rationalize
a numerator.
5
.
Ex: Rationalize the denominator of √3+4
√
Multiply the √top and bottom by the conjugate binomial 3 − 4:
5 √3−4
√5
= √3+4
.
3+4
3−4
We can
use the special
product
formula to simplify the denominator (it is of the form (x−y)(x+y)):
√
√
√
3−20
3−20
5
3−20
5
5
= (√3)2 +(−4)2 = 3−16 = −13
√
.
Ex: Rationalize the numerator: 3+x−4
x
√
Multiply by the conjugate 3 + x + 4:
√
3+x−4
x
√
√3+x+4
3+x+4
=
(3+x)−16
√
x 3+x+4x
=
√x−13
.
x 3+x+4x
Dividing Polynomials
To divide monomials, write the quotient as a fraction and simplify by using the rules of exponents.
Ex: Simplify:
=
9 3−1 2−3
x y
−3
9x3 y 2
−3xy 3
.
2
= −3x2 y −1 = − 3xy .
To divide a polynomial by a monomial, divide each term in the polynomial by the monomial.
Ex: Divide
5x3 y 2 +15x3 y 4
10x2 y 3
5x3 y 2 +15x3 y 4
and write
10x2 y 3
5x3 y 2
15x3 y 4
= 10x2 y3 + 10x2 y3
the answer without using negative exponents.
Notice that now we have a sum of monomials divided by monomials, so we can proceed as before:
x
= 2y
+ 3xy
.
2