MCR3U

Exam Review

Math Study Guide U.1: Rational Expressions, Exponents, Factoring, Inequalities

1.1 Exponent Rules Rule Product Quotient Power of a power Power of a product Power of a quotient

Description

a m × a n = a m+n a m ÷ a n = a m−n

Example 4 2 × 45 = 47

5 4 ÷ 52 = 52

(a )

a

m n

= a m×n

a a

(3 )

2 4

= 38

2 2 2

(xy) = x y an a = n ,b ≠ 0 b b

a0 = 1 a −m = 1 ,a ≠ 0 am

n

(2 x 3) = 2 x 3 35 3 = 5 4 4

70 = 1 9 −2 =

4 5

Zero as an exponent Negative exponents 1.2 Rational Exponents

1 92

a = a =

n m

power/root m/n

m n

( a)

n

m

m

27 3 = 3 274 =

(

3/2

3

27

)

4

a = a (alphabetical!) Negative Rational Exponents Rational = Fraction Radical = Root 1.3 Solving Exponential Equations e.g. Solve for x. -m/n m/n n -3/2 3/2 3/2

x = 1/x =1/ √x

(25/4) = (4/25) = (4 )/(25 ) =

3 3

√4 /√25 =8/125

9 x−2 − 8 = 73

Add 8 to both sides. Simplify.

x−2 = 2 x = 2+2 x=4

9 x−2 = 73 + 8 LS and RS are powers of Note 9 x−2 = 81 using the same base. 9 x −2 = 9 2 9, so rewrite them as powers

When the bases are the same, equate the exponents. Solve for x.

LS = 9 x−2 − 8 RS = 73 = 9 4− 2 − 8 = 81 − 8 = 73 = RS

Don’t forget to check your solution!

x = 4 checks

Exponential Growth and Decay Population growth and radioactive decay can be modelled using exponential functions.

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MCR3U

Decay:

t h

Exam Review

- initial amount t – time elapsed h – half-life A0

1 AL = A0 2

- amount Left

AL

Factoring Review Factoring Polynomials To expand means to write a product of polynomials as a sum or a difference of terms. To factor means to write a sum or a difference of terms as a product of polynomials. Factoring is the inverse operation of expanding. Expanding

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Sum or Product of difference polynomials of terms

Exam Review

( 2 x + 3)( 3x − 7 ) = 6 x 2 − 5x − 21

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Factoring Types of factoring:

Each term is divisible by .

Exam Review

e.g. Factor,

1+6x+9x2 is a perfect square trinomial Group 4mx – 4nx and ny – my, factor each group

Factor by grouping: group terms to help in the factoring process. e.g. Factor,

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Common of squares Recall n –factor Difference m = –(m – n)

Exam Review

A : 4mx + ny − 4nx − my = 4mx − 4nx + ny − my = 4 x( m − n ) + y ( n − m ) = 4 x( m − n ) − y ( m − n ) = ( 4 x − y )( m − n ) Factoring ax 2 + bx + c

B : 1 + 6x + 9x2 − 4 y2 = (1 + 3 x) 2 − 4 y 2 = [(1 + 3 x) + 2 y ][(1 + 3 x) − 2 y ] = (1 + 3 x + 2 y )(1 + 3 x − 2 y )

Find the product of ac. Find two numbers that multiply to ac and add to b. e.g. Factor,

A : y + 9 y + 14

2

Product = 14 = 2(7) Sum = 9 = 2 + 7

B : 3 x − 7 xy − 6 y

2

2

= y 2 + 7 y + 2 y + 14 = y ( y + 7) + 2( y + 7) = ( y + 2)( y + 7)

= 3x 2 − 9 xy + 2 xy − 6 y 2 = 3x( x − 3 y ) + 2 y ( x − 3 y ) = (3x + 2 y )( x − 3 y )

Product = 3(–6) = –18 = –9(2) Sum = – 7 = –9 + 2 Decompose middle term – 7xy into –9xy + 2xy. Factor by grouping.

Sometimes polynomials can be factored using special patterns. Perfect square trinomial or a 2 + 2ab + b 2 = (a + b)(a + b) e.g. Factor, a 2 − 2ab + b 2 = (a − b)(a − b)

A : 4 p 2 + 12 p + 9 = (2 p + 3) 2

B : 100x 2 − 80xy + 16 y 2 = 4(25x 2 − 20xy + 4 y 2 ) = 4(5 x − 2 y )(5 x − 2 y )

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MCR3U

Difference of squares

Exam Review

a 2 − b 2 = (a + b)(a − b)

e.g. Factor, 9 x 2 − 4 y 2 = (3 x + 2 y )(3x − 2 y ) 1.4 +, - , X, Polynomials A polynomial is an algebraic expression with real coefficients and non-negative integer exponents. A polynomial with 1 term is called a monomial, . 7x A polynomial with 2 terms is called a binomial, 3x − 9

2

. .

3x + 7 x − 9

2

A polynomial with 3 terms is called a trinomial,

The degree of the polynomial is determined by the value of the highest exponent of the variable in the polynomial. e.g. , degree is 2. 3x 2 + 7 x − 9

For polynomials with one variable, if the...