Mathematics
Grade 9
15 min
Factor out a monomial
Factor out a monomial
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define monomial, polynomial, and Greatest Common Factor (GCF).
Identify the GCF of the terms in a given polynomial.
Factor out a monomial GCF from a binomial.
Factor out a monomial GCF from a trinomial or other polynomial.
Use the distributive property to verify their factored answer.
Recognize when a polynomial is fully factored by removing the GCF.
Ever tried to reverse a math problem? 🤔 Factoring is like playing the distributive property in reverse, and it's a key that unlocks more advanced algebra!
In this tutorial, you will learn how to identify and 'pull out' the greatest common factor (GCF) from a polynomial. This is the first and most important step in factoring, a fundamental skill used for simplifying expressions, solving quadra...
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Key Concepts & Vocabulary
TermDefinitionExample
MonomialA single algebraic term consisting of a number, a variable, or a product of numbers and variables with whole-number exponents.7, 5x, -3a^2b
PolynomialAn algebraic expression consisting of one or more monomials (terms) connected by addition or subtraction.4x^2 + 8x, 9y^3 - 3y + 1
Factor (noun)A number or expression that divides another number or expression evenly, with no remainder.The factors of 12 are 1, 2, 3, 4, 6, and 12.
Factor (verb)The process of breaking down a number or expression into a product of its factors.To factor 2x + 4 is to write it as 2(x + 2).
Greatest Common Factor (GCF)The largest monomial that is a factor of every term in a polynomial.The GCF of 15x^3 and 25x^2 is 5x^2.
Distributive PropertyThe rule that states a(b + c) = ab + ac. Facto...
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Core Formulas
The Factoring Process
Polynomial = GCF * (Term1/GCF + Term2/GCF + ...)
This is the general procedure. First, find the GCF of all terms. Second, write the GCF outside a set of parentheses. Third, divide each original term by the GCF to find the new terms inside the parentheses.
Reverse Distributive Property
ab + ac = a(b + c)
This is the fundamental principle behind factoring. The common factor 'a' is 'pulled out' from each term, leaving the remaining parts 'b' and 'c' inside the parentheses.
Exponent Rule for Division
x^m / x^n = x^(m-n)
When finding the GCF of variables, you take the one with the lowest exponent. This rule is then used to divide the variables in each term by the GCF's variable part.
4 more steps in this tutorial
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Challenging
The area of a rectangular garden is represented by the polynomial 24x^5y^3 - 16x^3y^4 square meters. If the width of the garden is represented by the greatest common monomial factor of the terms, what is the length?
A.3x^2 - 2y
B.8x^3y^3
C.3x^2y - 2xy^2
D.8x^2(3x^3y^3 - 2xy^4)
Challenging
If the GCF of the two terms in the polynomial ax^m + bx^n is 7x^4, where m > n, which of the following statements MUST be true?
A.n = 4, and both a and b are multiples of 7.
B.m = 4, and a is a multiple of 7.
C.n = 4, and a = 7.
D.m > 4, and b is a multiple of 7.
Challenging
Factor the polynomial completely: -x^3y^2 - x^2y^3 - xy^4
A.-xy(x^2y + xy^2 + y^3)
B.-x^2y^2(x + y)
C.-xy^2(x^2 + xy + y^2)
D.xy^2(-x^2 - xy - y^2)
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