Mathematics
Grade 11
15 min
Factor out a monomial
Factor out a monomial
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the Greatest Common Factor (GCF) of the terms in a polynomial.
Determine the monomial GCF for polynomials containing multiple variables and exponents.
Factor out a monomial GCF from binomials, trinomials, and other polynomials.
Correctly handle sign changes when factoring out a negative monomial.
Apply the distributive property in reverse to express a polynomial as a product of a monomial and another polynomial.
Verify their factored result by multiplying the factors to obtain the original polynomial.
Ever look at a complex equation and wish you could simplify it into something more manageable? 🤔 Factoring is the first step to making that happen!
This tutorial focuses on a fundamental algebra skill: factoring out a monomial. This process is ess...
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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 non-negative integer exponents.`7x^3y` is a monomial. `7x + 3y` is not.
PolynomialAn algebraic expression consisting of one or more monomials (terms) combined by addition or subtraction.`12a^4 - 5a^2 + 3` is a polynomial.
Factor (verb)The process of breaking down a polynomial into a product of simpler expressions (its factors).To factor `6x + 18` is to rewrite it as `6(x + 3)`.
Greatest Common Factor (GCF)The largest monomial that divides into each term of a polynomial without a remainder. It includes the GCF of the numerical coefficients and the GCF of the variable parts.The GCF of `10x^3` and `25x^2` is `5x^2`.
Distributive PropertyThe rule stating that `a...
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Core Formulas
The Reverse Distributive Property
ab + ac = a(b + c)
This is the fundamental principle of factoring out a monomial. The common factor 'a' is 'pulled out', and the remaining parts of each term, 'b' and 'c', are left inside the parentheses.
Rule for GCF of Variables
For terms `x^m, x^n, ...`, the GCF is `x^k` where `k = min(m, n, ...)`
To find the GCF of variable parts, identify the common variables and for each one, choose the lowest exponent that appears in any of the terms.
The Factoring Process
Polynomial = GCF * (Term1/GCF + Term2/GCF + ...)
First, find the GCF of all terms. Then, write the GCF outside a set of parentheses. Inside the parentheses, write the result of dividing each original term by the GCF.
4 more steps in this tutorial
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Challenging
When finding the GCF of variable terms like `x^m, x^n, ...`, the rule is to use `x^k` where `k = min(m, n, ...)`. Why is the minimum exponent used?
A.It is a mathematical convention with no specific reason.
B.Because the GCF must be able to divide into every term, and a higher power would not divide into a term with a lower power.
C.Because the maximum exponent represents the least common multiple (LCM), so the minimum must be for the GCF.
D.To ensure the remaining polynomial factor has only positive exponents.
Challenging
Factor the polynomial whose terms are not in standard order: `30m^2np^3 - 50m^3n^2p + 10m^2n`.
A.10m^2(3np^3 - 5mn^2p + n)
B.10mn(3mp^3 - 5m^2np + m)
C.10m^2n(3p^3 - 5mp + 1)
D.5m^2n(6p^3 - 10mp + 2)
Challenging
The surface area of a cylinder is given by `2Ï€r^2 + 2Ï€rh`. What is this formula in factored form?
A.2Ï€r(r + h)
B.2Ï€(r^2 + rh)
C.2r(πr + πh)
D.Ï€r(2r + 2h)
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