Factoring (chap 5)
1. x 2+ bx + c, c positive
In this section, you will study trinomials that can be factored as a product (x + r)(x + s), where are both positive or both negative integers. The diagram below shows that the product (x + r)(x + s) and the trinomial
x2 + (r+s)x + rs represent the same total area. Note that the coefficient of the x-term is the sum of r and s, and the constant term is the product.
2. x2 + bx + c, c negative
In the last section, the variable c was positive which meant that r and s were both positive or negative. In this section, c negative which means r and s are opposite signs. The method used in the section before is still the same though.
3. ax2 +bx + c
If ax2 + bx + c (a > 1) can be factored, the factorization will have the pattern (px + r)(qx + s)
Example 1. Factor 2x2
Solution
Clue 1- Because the trinomial has a negative constant, one of either r or s has to be negative.
Clue 2- List the possible factors of the constant and quadratic term.
Test the possibilities to see which produces the correct term, 7x. Making a chart will help you.
4. Difference of Squares
Difference of two squares is in the equation (a + b)(a - b). When there is this equation the answer is a2 - b2.
(a + b)(a - b) = a2 - b2
(sum of two numbers) times (their difference) = (first number)2 - (second number)2
5. Perfect Square Trinomial
A perfect square trinomial is when the factors that come from factoring the equation. The answer will be the exact same thing.
(a + b)2 = a2 + 2ab + b2
Example 1
example 2
6. Guidelines for Factoring Completely
1. Factor out the greatest monomial factor first.
2. Look for a difference of two squares.
3. Look for a perfect square trinomial.
4. If a trinomial is not a square, look for a pair of binomial factors.
5. Check to make sure that all binomial or trinomial factors are prime.
6. Multiply the factors to check your work.
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