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Day 3: Factoring Polynomials (GCF)
To be able to factor polynomials by finding and dividing out the Greatest Common Factor (GCF).
Yesterday, we learned how to multiply. Today, we learn how to do the reverse!
\(5(x + 3) \iff 5x + 15\)
Factoring is the process of finding what you multiplied to get an expression.
The Greatest Common Factor is the largest number and/or variable that divides evenly into all terms of the polynomial.
How to Find the GCF:
Let's find the GCF of \(12x^3 + 18x^2\).
Coefficients: The GCF of 12 and 18 is 6.
Variables: We have \(x^3\) and \(x^2\). The one with the lowest exponent is \(x^2\). The GCF is \(x^2\).
The GCF of the whole polynomial is \(6x^2\).
Now, let's factor \(12x^3 + 18x^2\).
// Step 1: We found the GCF is \(6x^2\). Put it outside the parentheses.
\(6x^2(\quad ? \quad + \quad ? \quad)\)
// Step 2: Divide each original term by the GCF.
\(\frac{12x^3}{6x^2} = 2x \quad \text{and} \quad \frac{18x^2}{6x^2} = 3\)
// Step 3: Put the results inside the parentheses.
\(6x^2(2x + 3)\)
Check your answer: Distribute \(6x^2(2x+3)\) to see if you get back to \(12x^3 + 18x^2\)!
Let's factor this trinomial as a class.
\(8a^3 - 4a^2 + 12a\)
Hint: What is the GCF of 8, -4, and 12? What is the GCF of \(a^3, a^2,\) and \(a\)?
Factor the following polynomial completely.
\(5y^3 - 10y^2\)
\(5y^2(y - 2)\)
Try these problems on your handout, starting with green.
Green Level
FACTOR: \(4x + 8\)
Yellow Level
FACTOR: \(3n^2 - 9n + 15\)
Red Level
FACTOR: \(14p^4 - 21p^3 + 7p^2\)
On your handout or a piece of paper, factor the following expression completely: