Download Algebraic Expressions (continued)

Algebraic Expressions (continued)
Factoring
Examples: Factor each expression
−2x3 + 16x
2x2 y − 6xy 2 + 3xy
To factor a polynomial of the form x2 + bx + c, we need to find two numbers
that add up to b and multiply to c.
x2 − 6x + 5
To factor a polynomial of the form ax2 + bx + c, we need to find factors
(px + r) and (qx + s) such that
ax2 + bx + c = (px + r)(qx + s) = pqx2 + (ps + qr)x + rs
That means we look for numbers p, q, r and s such that pq = a, ps + qr = b
and rs = c.
3x2 − 16x + 5
3(3y + 2)2 − 16(3y + 2) + 5
1
Special factoring formulas
A2 − B 2 = (A − B)(A + B)
A2 + 2AB + B 2 = (A + B)2
A2 − 2AB + B 2 = (A − B)2
A3 − B 3 = (A − B)(A2 + AB + B 2 )
A3 + B 3 = (A + B)(A2 − AB + B 2 )
Examples:
16z 2 − 24z + 9
1 + 1000y 3
More factoring examples:
Factoring by grouping terms.
x3 + x2 + x + 1
Factoring expressions with rational exponents.
x−3/2 + 2x−1/2 + x1/2
Factor completely.
y 4 (y + 2)3 + y 5 (y + 2)4
2
difference of squares
perfect square
perfect square
difference of cubes
sum of cubes
Rational Expressions
A quotient of two polynomials is called a rational expression. A quotient of
two algebraic expression is called a fractional expression.
√
x5 + 2
x2 − 1
x−4
+ x3
x3/2
The domain of a quotient is the set of real numbers that the variable in the
quotient is permitted to have, so that the quotient is defined.
Examples:
Find the domain of the following quotients.
x2 + 1
x2 − x − 2
√
2x
x+1
How to handle fractional expressions:
We use the laws of inverses of real numbers (Lecture 1) to simplify, multiply,
divide, add or subtract fractional expressions.
Examples:
Perform the multiplication, division, addition or subtraction if needed and
simplify.
x2 − x − 12
x2 + 5x + 6
3
1 − x2
x3 − 1
x2 − 2x − 15 x + 3
·
x2 − 9
x−5
x3
x+1
x
x2 +2x+1
3
x
−
x−4 x+6
2
3
1
−
+ 2
2
x + 1 (x + 1)
x −1
Simplify the following compound fraction .
x
y
1
x2
−
−
y
x
1
y2
4
x−2 − y −2
x−1 + y −1
1
1+x+h
−
h
1
1+x
Rationalizing
the denominator (numerator) means to eliminate terms like
√
both the numerA + C from the denominator (numerator) by multiplying
√
ator and the denominator by the conjugate A − C.
Examples:
1
√
2− 3
√
√
x− x+h
√
h+ x+h
5