Outro W9

Revie of the week

You can find the key concepts you studied this week in the following short reviews

7.1 Greatest Common Factor and Factor by Grouping

Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Step 3. Bring down the common factors that all expressions share.
4. Step 4. Multiply the factors as in Example 7.2 Links to an external site..
Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
1. Step 1. Find the GCF of all the terms of the polynomial.
2. Step 2. Rewrite each term as a product using the GCF.
3. Step 3. Use the ‘reverse’ Distributive Property to factor the expression.
4. Step 4. Check by multiplying the factors as in Example 7.5 Links to an external site..
Factor by Grouping: To factor a polynomial with 4 four or more terms
1. Step 1. Group terms with common factors.
2. Step 2. Factor out the common factor in each group.
3. Step 3. Factor the common factor from the expression.
4. Step 4. Check by multiplying the factors as in Example 7.15 Links to an external site..

7.3 Factor Trinomials of the Form $ax^2+bx+c$

Factor Trinomials of the Form $ax^2+bx+c$ using Trial and Error: See Example 7.33 Links to an external site..
1. Step 1. Write the trinomial in descending order of degrees.
2. Step 2. Find all the factor pairs of the first term.
3. Step 3. Find all the factor pairs of the third term.
4. Step 4. Test all the possible combinations of the factors until the correct product is found.
5. Step 5. Check by multiplying.
Factor Trinomials of the Form $ax^2+bx+c$ Using the “ac” Method: See Example 7.38 Links to an external site..
1. Step 1. Factor any GCF.
2. Step 2. Find the product ac.
3. Step 3. Find two numbers m and n that:
  $\begin{matrix} Multiply to a c & m \cdot n = a \cdot c \\ Add to b & m + n = b \end{matrix}$ m⋅n=a⋅c

$\begin{matrix} Multiply to a c & m \cdot n = a \cdot c \\ Add to b & m + n = b \end{matrix}$

1. Step 4. Split the middle term using m and n:
2. Step 5. Factor by grouping.
3. Step 6. Check by multiplying the factors.
Choose a strategy to factor polynomials completely (updated):
1. Step 1. Is there a greatest common factor? Factor it.
2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?
  If it is a binomial, right now we have no method to factor it.
  If it is a trinomial of the form $x^2+bx+c$
     Undo FOIL $(x) (x)$ .
  If it is a trinomial of the form $ax^2+bx+c$
     Use Trial and Error or the “ac” method.
  If it has more than three terms
     Use the grouping method.
3. Step 3. Check by multiplying the factors.

7.4 Factor Special Products

Factor perfect square trinomials See Example 7.42 Links to an external site..
$\begin{matrix} Step 1. Does the trinomial fit the pattern? & a^{2} + 2 a b + b^{2} & a^{2} - 2 a b + b^{2} \\ Is the first term a perfect square? & {(a)}^{2} & {(a)}^{2} \\ Write it as a square. \\ Is the last term a perfect square? & {(a)}^{2} {(b)}^{2} & {(a)}^{2} {(b)}^{2} \\ Write it as a square. \\ Check the middle term. Is it 2 a b ? & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} \\ Step 2. Write the square of the binomial. & {(a + b)}^{2} & {(a - b)}^{2} \\ Step 3. Check by multiplying. \end{matrix}$

$\begin{matrix} Step 1. Does the trinomial fit the pattern? & a^{2} + 2 a b + b^{2} & a^{2} - 2 a b + b^{2} \\ Is the first term a perfect square? & {(a)}^{2} & {(a)}^{2} \\ Write it as a square. \\ Is the last term a perfect square? & {(a)}^{2} {(b)}^{2} & {(a)}^{2} {(b)}^{2} \\ Write it as a square. \\ Check the middle term. Is it 2 a b ? & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} \\ Step 2. Write the square of the binomial. & {(a + b)}^{2} & {(a - b)}^{2} \\ Step 3. Check by multiplying. \end{matrix}$

Factor differences of squares See Example 7.47 Links to an external site..
$\begin{matrix} Step 1. Does the binomial fit the pattern? & a^{2} - b^{2} \\ Is this a difference? & ____-____ \\ Are the first and last terms perfect squares? \\ Step 2. Write them as squares. & {(a)}^{2} - {(b)}^{2} \\ Step 3. Write the product of conjugates. & (a - b) (a + b) \\ Step 4. Check by multiplying. \end{matrix}$

$\begin{matrix} Step 1. Does the binomial fit the pattern? & a^{2} - b^{2} \\ Is this a difference? & ____-____ \\ Are the first and last terms perfect squares? \\ Step 2. Write them as squares. & {(a)}^{2} - {(b)}^{2} \\ Step 3. Write the product of conjugates. & (a - b) (a + b) \\ Step 4. Check by multiplying. \end{matrix}$

Factor sum and difference of cubes To factor the sum or difference of cubes: See Example 7.54 Links to an external site..
1. Step 1. Does the binomial fit the sum or difference of cubes pattern? Is it a sum or difference? Are the first and last terms perfect cubes?
2. Step 2. Write them as cubes.
3. Step 3. Use either the sum or difference of cubes pattern.
4. Step 4. Simplify inside the parentheses
5. Step 5. Check by multiplying the factors.

7.5 General Strategy for Factoring Polynomials

General Strategy for Factoring Polynomials See Figure 7.4 Links to an external site..
How to Factor Polynomials
1. Step 1. Is there a greatest common factor? Factor it out.
2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?
  - If it is a binomial:
    Is it a sum?
    - Of squares? Sums of squares do not factor.
    - Of cubes? Use the sum of cubes pattern.
    Is it a difference?
    - Of squares? Factor as the product of conjugates.
    - Of cubes? Use the difference of cubes pattern.
  - If it is a trinomial:
    Is it of the form $x^2+bx+c$ ? Undo FOIL.
    Is it of the form $ax^2+bx+c$ ?
    - If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
    - Use the trial and error or ‘ac’ method.
  - If it has more than three terms:
    Use the grouping method.
3. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

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