MATH REVIEWER 1st QUARTER
HOW TO KNOW IF THE GIVEN IS A PERFECT SQUARE
TRINOMIAL
SPECIAL PRODUCTS
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1. Find the square root of the first term
2. Find the square root of the last term/third term
3. Multiply the square root of the first term with
the last term, then by 2
4. If the answer is correct, then it is a perfect
square trinomial
Are products that occur frequently in algebra
and have patterns which make them easy to
obtain
POLYNOMIALS WITH COMMON MONOMIAL FACTOR
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One type of special product are the polynomials
with common monomial factor
PRIME POLYNOMIAL
Any polynomial whose greatest common
monomial factor is 1 is described as prime
polynomial
A prime polynomial has only two factors; 1 and
the polynomial itself
TRINOMIALS THAT ARE SQUARE OF BINOMALS
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SQUARE OF A BINOMIAL
(𝑥 + 𝑦)2 = (𝑥 + 𝑦)(𝑥 + 𝑦)
= 𝑥 2 + 2𝑥𝑦 + 𝑦 2
(𝑥 − 𝑦)2 = (𝑥 − 𝑦)(𝑥 − 𝑦)
= 𝑥 2 − 2𝑥𝑦 + 𝑦 2
SQUARE OF A MULTINOMIAL
(𝑎𝑥 + 𝑏)2 = 𝑎2 𝑥 2 + 2𝑎𝑏𝑥 + 𝑏 2
A trinomial that results from squaring a
binomial is said to be a perfect square trinomial
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TRINOMIALS THAT ARE PRODUCT OF TWO BINOMIALS
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The square of a multinomial is equal to the sum
of the squares of each term, plus twice the
product of all possible combinations of all the
terms taken two at a time
EXAMPLE:
(𝑎𝑥 + 𝑏𝑦)(𝑐𝑥 + 𝑑𝑦) = 𝑎𝑐𝑥 2 + (𝑏𝑐 + 𝑎𝑑)𝑥𝑦 + 𝑏𝑑𝑦 2
(𝑎 + 𝑏 + 𝑐)2 = (𝑎 + 𝑏 + 𝑐)(𝑎 + 𝑏 + 𝑐)
BINOMIALS THAT ARE PRODUCT OF A SUM AND
DIFFERENCE OF TWO TERMS
= 𝑎2 +𝑏 2 + 𝑐 2 + 2𝑎𝑏 + 2𝑎𝑐 + 2𝑏𝑐
1. (𝑎𝑥 + 𝑏𝑦)(𝑎𝑥 − 𝑏𝑦) = 𝑎2 𝑥 2 − 𝑏 2 𝑦 2
2. (𝑥 + 𝑦)(𝑥 2 − 𝑥𝑦 + 𝑦 2 ) = 𝑥 3 +𝑦 3
3. (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦 2 ) = 𝑥 3 − 𝑦 3
FACTORING
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EXAMPLES:
Is the reverse process of multiplication
COMMON MONOMIAL FACTORS
COMMON MONOMIAL
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Polynomial
Factors
Special Products
3x + 7
2
6x+14 GCF:2
2 + 8𝑥𝑦 GCF:2x
6𝑥
3x + 4y
2x
2
2
16𝑥 2 𝑦 3 + 12𝑦 2 GCF: 4y
4𝑦
4𝑥 𝑦 + 3
TRINOMIALS THAT ARE SQUARES OF BINOMIALS
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2
(𝑎𝑥 + 𝑏)2 = (𝑎𝑥 + 𝑏)(𝑎𝑥 + 𝑏)
= 𝑎2 𝑥 2 + 2𝑎𝑏𝑥 + 𝑏 2
(2𝑥 + 5)2 = 4𝑥 2 + 20𝑥 + 25
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When a number or a polynomial is factored, it is
rewritten as a product of two or more factors
A polynomial said to be factored into prime
factors if it is expressed as the product of two or
more irreducible polynomials of the same type
A polynomial is factored completely if each of
its factors can no longer be expressed as
product of two other polynomials of lower
degree and that the coefficient have no
common factors
GREATEST COMMON FACTORS
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The greatest common factor (GCF) is the largest
number that a set of numbers or polynomials
have in common.
EXAMPLE:
(𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2
(𝑥 − 𝑦)2 = 𝑥 2 − 2𝑥𝑦 + 𝑦 2
PST: 𝑥 2 + 2𝑥𝑦 + 𝑦 2 = (𝑥 + 𝑦)2
EXAMPLE:
(𝑥 + 𝑦)(2𝑎) + (𝑥 + 𝑦)(𝑏) = (𝑥 + 𝑦)(2𝑎 + 𝑏)
√𝑥 2 = 𝑥
COMMON MONOMIAL FACTORING
√𝑦 2 = 𝑦
𝑥 ∙ 𝑦 ∙ 2 = 2𝑥𝑦
1. Find the greatest common (GCF) of the terms in
the polynomial. This is the first factor
2. Divide each terms by the GCF to get the other
factor
- Grouping the terms in a polynomial is also a
useful technique in factoring. Some polynomials
can be factored by grouping terms in such a
way as to get polynomials with special factors.
FACTORS OF DIFFERENCE OF TWO SQUARES
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The product of sum and difference of two
numbers was the difference between two
squares
FACTORING THE DIFFERENCE OF TWO SQUARES
1. Get the principal square root of each of the two
square
2. Using these square roots, form two factors, one
is sum, and the other is difference
FACTORS OF SUM AND DIFFERENCE OF TWO CUBES
Factoring a Sum of Cubes:
a3 + b3 = (a + b)(a2 – ab + b2)
Factoring a Difference of Cubes:
a3 – b3 = (a – b)(a2 + ab + b2)
STEPS IN FACTORING THE SUM OR DIFFERENCE OF
TWO CUBES
1. Get the cube root of each cube
2. Taking the operation between the cubes,
obtain a binomial factor using the cube roots
in steep 1
3. From the second trinomial factor as follows
a. Square the first two cube root
b. Multiply the two cube roots
EXAMPLE:
c. Square the second cube root
2
(𝑥 + 𝑦)(𝑥 − 𝑦) = 𝑥 − 𝑦
2
𝑥 3 −𝑦 3 = (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦 2 )
𝑥 2 − 𝑦 2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
3√𝑥 3 = 𝑥
√𝑥 2 = 𝑥
3√𝑦 3 = 𝑦
√𝑦 2 = 𝑦
FRACTIONS
FACTORS OF THE PERFECT SQUARE TRINOMIAL
FACTORING PERFECT SQUARE TRINOMIALS (PST)
1. Get the square roots of the first and last terms
2. Use the sign of the middle term between these
roots
3. Square the binomial obtained in step 2
𝑎
A rational number in the form 𝑏 , where 𝑏 ≠ 0
is called a fraction
SIMPLIFYING FRACTIONS
1. Find the GCF
2. Divide the numerator and denominator to the
GCF
OPERATIONS
MULTIPLICATION
1.
Multiply the numerator by the numerator.
2.
Multiply the denominator by the denominator.
COMPLEX FRACTIONS
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Have fractions on their numerator or
denominator, or in both
SIMPLIFYING COMPLEX FRACTIONS
BY COMBINING TERMS
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DIVISION
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Multiply by its reciprocal
BY MULTIPLYING THE LCD
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Adding and Subtracting Fractions with Like
Denominators
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Express the numerator and denominator of the
complex fraction as simple fractions. Then
divide the fractions and simplify
Multiply the numerator and denominator of the
complex fraction by the LCD of the minor
denominators
RATIONAL ALGEBRAIC EXPRESSION
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Just add or subtract the numerators, and write
the result over the same denominator.
Is nothing more than a fraction in which the
numerator and/or the denominator are
polynomials.
SIMPLIFYING RATIONAL EXPRESSION
Adding and Subtracting Fractions with Unlike
Denominators
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If the denominators are not the same, then
you have to use equivalent fractions which
do have a common denominator. To do this,
you need to find the least common multiple
(LCM) of the two denominators.
To simplify a rational expression you have to
eliminate all factors that are common of the
numerator and the denominator. To accomplish
this use the greatest common factor (GCF) of
the factors e.g.