Factored Form

Quadratic Relations

Learning Goals for Unit

  • Factor Binomials and Trinomials
  • Use factored form to find AOS

Factored Form: a(x-r)(x-s)

  1. The value of a gives you the shape and direction of opening
  2. The value of r and s are the 2 x-intercepts (s,0) and (r,0)
  3. To find the y-intercept (0,y) set x=0 then solve for y
  4. To find the axis of symmetry, add both x-intercepts and divide by 2
  5. To find the vertex, use the zeros (x-intercepts) to find the AOS and sub this x value into the equation to solve for y.

Graphing Factored Form

Graphing a Quadratic Function in Factored Form

Expanding and Simplifying

The distributive property of multiplication remembering this will make it very easy to multiply binomials. You may be wondering what is a binomial? A binomial is a polynomial with two terms.
Multiply Binomials

Types of Factoring

Monomial Common Factor:

To find the GCF of the following expression, find the GCF of the coefficients and the GCF of the variables. As you can see from the example.

This means you must find a number that is a multiple of all numbers in the equation, also find a variable present in all numbers and put it outside the brackets.

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Binomial Common Factors:

If there are two binomials that are exactly the same, think of the Binomial as one factor.
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Factoring by Grouping:

Group the 2 terms that have a common factor.

Then perform the same steps used for monomial factoring.

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Factoring Trinomials (from Standard form) a^2+bx+c

In the standard form, r+s=b and rs+c. R and s are integers


So this means you must find 2 number who's sum is B and the product is C.

You then take factors of b, input them into the equation, removing b but keeping the variables.

Then factor by grouping, for both groups the numbers inside the brackets should be the same. You then take the numbers outside the brackets and put them together.

Factoring Complex Trinomials

Complex trinomials have a coefficient in front of "a"

To solve this, you multiply c by the coefficient of a, use this new number and perform the steps used in regular trinomial factoring while keeping c the same in the equation.

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Special Product: Difference of squares

Two terms that are squared and separated by a subtraction sign like this:

a2 - b2

Useful because it can be factored from(a+b)(a−b) to a^2-b^2
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Special Product: Perfect square trinomials

In the picture below, it describes how special case perfect trinomials work.
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Application Problems

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2(x² +2x -168) = 0

(Factor)

(x+14)(x-12)


14² + 12² = 340

196+ 144+340


Therefore the integers are 12 and 14

Factoring Trinomials

factoring trinomials