essential question: how is foil related to factoring?

Essential Question: How is FOIL related to factoring?

5-4: Factoring Quadratic ExpressionsQuadratic Functions (you saw this in 5-1)

A quadratic function is one whose largest term uses x2

It’s written in standard form asf(x) = ax2 + bx + c a, b, and c represent coefficients (real numbers) The x2 terms comes first, followed by the x term,

followed by the term that doesn’t have an x The x2 term and x term cannot be combined

5-4: Factoring Quadratic ExpressionsFOIL (Note: You saw this in 5-1)

FOIL is an acronym for “First, Outer, Inner, Last” Multiply the indicated terms together Combine like terms

Example: y = (2x + 3)(x – 4)

y = (2x + 3)(x – 4)

FirstLast

Inner

Outer

First: 2x • x = 2x2

Outer: 2x • -4 = -8xInner: 3 • x = 3xLast: 3 • -4 = -12

y = 2x2 – 8x + 3x – 12y = 2x2 – 5x - 12

5-4: Factoring Quadratic ExpressionsFOIL

(x – 4)(x + 3)

(-x – 5)(3x – 1)

x2 – 4x + 3x – 12x2 – x – 12

-3x2 – 15x + x + 5-3x2 – 14x + 5

5-4: Factoring Quadratic ExpressionsFinding the Greatest Common Factor (GCF)

The GCF of an expression is the common factor with the greatest coefficient and the smallest exponent

Example: Factor 4x6 + 20x3 – 12x2

The largest coefficient that can divide 4, 20, and -12 is 4

The smallest exponent is x2 4x2(x4) + 4x2(5x) + 4x2(-3) 4x2(x4 + 5x – 3)

5-4: Factoring Quadratic ExpressionsFactor

4w2 + 2w

5t4 + 7t2

GCF: 2wFactored: 2w(2w + 1)

GCF: 1t2

Factored: t2(5t2 + 7)

5-4: Factoring Quadratic ExpressionsAssignment

FOIL/GCF worksheetDo all problemsShow your work

5-4: Factoring Quadratic ExpressionsFactoring: The steps (Holy Grail algorithm)

In standard form: f(x) = ax2 + bx + c1.Find two numbers with:

A product of a • c A sum of b

2.Use those two numbers to split the “b” term3.Factor out the GCF from the first two terms as well

as the last two terms4.You know you’ve factored correctly if both binomials

inside the parenthesis match5.Combine the terms outside parenthesis into their

own parenthesis

+

5-4: Factoring Quadratic ExpressionsSome hints (summarized on next slide):

The a term should be positive (I won’t give you otherwise) If not, flip the signs on each term -x2 + 5x + 24 gets flipped into x2 – 5x – 24

If a • c is positive, the two numbers you’re looking for are going to be the same sign as b ex #1) x2 + 9x + 20 4 & 5 ex #2) x2 – 11x + 28 -4 & -7

Why? Because only a positive • positive and/or negative • negative = positive

If a • c is negative, the bigger of the two numbers will have the same sign as b ex #3) x2 + 3x – 10 5 & -2 ex #4) x2 – 5x – 24 -8 & 3

Why? Because only a negative • positive = negative

5-4: Factoring Quadratic Expressions

Multiply:+ number

Multiply:- number

Add:+ number

Add:- number

Add:+ number

Add:- number

Both #s are + Both #s are - Bigger # is + Bigger # is -

Some hints about finding the two numbers to be used in factoring:

5-4: Factoring Quadratic ExpressionsFactoring (Example #4)

Factor: 3x2 – 16x + 5 a = 3, c = 5 → ac = 15 Find two numbers that:

multiply together to get 15 add to get -16

Possibilities: -1/-15, -3/-5Rewrite the b term

3x2 – 1x – 15x + 5Factor GCF from first two and last two terms

x(3x – 1) – 5(3x – 1)Combine terms outside the parenthesis

(x – 5)(3x – 1)

3x2 – 16x + 5

+

-1 -153x2 x x + 5x(3x – 1) -5(3x – 1)

(x – 5)(3x – 1)


2x2 + 11x + 12


Factor: 4x2 – 4x – 15 a = 4, c = -15 → ac = -60 Find two numbers that:

multiply together to get -60(1 positive, 1 negative)

add to get -4 (larger is negative) Possibilities: 1/-60, 2/-30, 3/-20,

4/-15, 5/-12, 6/-10Rewrite the b term

4x2 + 6x – 10x – 15Factor GCF from first two and last two terms

2x(2x + 3) – 5(2x + 3)Combine terms outside the parenthesis

(2x – 5)(2x + 3)

4x2 – 4x – 15

+

+6 -104x2 x x – 152x(2x + 3) -5(2x + 3)

(2x – 5)(2x + 3)


6x2 + 11x – 35


Pg. 26325 – 36 (all problems)No work = no credit

Additional examples (and steps) are available at http://www.gushue.com/factoring2.php

5-4: Factoring Quadratic ExpressionsFactoring: The steps (same as last week)

In standard form: f(x) = ax2 + bx + cFind two numbers with:

A product of a • c A sum of b

Use those two numbers to split the “b” termFactor out the GCF from the first two terms as well

as the last two termsYou know you’ve factored correctly if both binomials

inside the parenthesis matchCombine the terms outside parenthesis into their

own parenthesis


Factor: x2 + 8x + 7 a = 1, c = 7 → ac = 7 Find two numbers that:

multiply together to get 7 add to get 8

Only possibility is 1/7Rewrite the b term

x2 + 1x + 7x + 7Factor GCF from first two and last two terms

x(x + 1) + 7(x + 1)Combine terms outside the parenthesis

(x + 7)(x + 1)

x2 + 8x + 7

+

+1 +7x2 x x + 7x(x + 1) +7(x + 1)

(x + 7)(x + 1)

5-4: Factoring Quadratic ExpressionsYour Turn. Factor:

x2 + 4x – 5

x2 – 12x + 11

Two numbers? 5 & -1x2 + 5x – 1x – 5x(x + 5) -1(x + 5)(x – 1)(x + 5)

Two numbers? -11 & -1x2 – 11x – 1x + 11x(x – 11) -1(x – 11)(x – 1)(x – 11)


Multiply:+ number

Multiply:- number

Add:+ number

Add:- number

Add:+ number

Add:- number

Both #s are + Both #s are - Bigger # is + Bigger # is -

Some hints about finding the two numbers to be used in factoring:


Factor: x2 – 17x + 72 a = 1, c = 72 → ac = 72 Find two numbers that:

multiply together to get 72 (both + or both –) add to get -17 (both –)

Possibilities: -1/-72, -2/-36, -3/-24, -4/-18, -6/-12, -8/-9Rewrite the b term

x2 – 8x – 9x + 72Factor GCF from first two and last two terms

x(x – 8) + -9(x – 8)Combine terms outside the parenthesis

(x – 9)(x – 8)


x2 + 8x + 15

x2 – 5x + 6Two numbers? -2 & -3x2 – 2x – 3x + 6x(x – 2) -3(x – 2)(x – 3)(x – 2)

Two numbers? 3 & 5x2 + 3x + 5x + 15x(x + 3) +5(x + 3)(x + 5)(x + 3)


x2 + 4x – 12

x2 – 2x – 15

Two numbers? -2 & 6x2 – 2x + 6x – 12x(x – 2) +6(x – 2)(x + 6)(x – 2)

Two numbers? 3 & -5x2 + 3x – 5x – 15x(x + 3) -5(x + 3)(x – 5)(x + 3)


Pg. 2637 – 24 (all problems)

Additional examples (and steps) are available at http://www.gushue.com/factoring.php

5-4: Factoring Quadratic ExpressionsThere are two special cases to discuss:

The Difference of Perfect Squares x2 – 16 If we’re using the Holy Grail Algorithm:

a = 1 b = 0 (there’s no ‘x’ term) c = -16

So we’re looking for two numbers that multiply to get -16 (1 • -16) and add together to get 0

The only way to have two numbers that add together to get 0 is if they’re opposites, in this case 4 & -4

5-4: Factoring Quadratic ExpressionsFactoring: x2 - 16

x2 + 0x – 16

+

-4 +4x2 x x – 16x(x – 4) +4(x – 4)

(x + 4)(x – 4)


9x2 – 25

The shortcut: Take the square root of the left term: Take the square root of the right term: Write the factor as a sum and difference of the squares

3x5

(3x + 5)(3x – 5)

5-4: Factoring Quadratic ExpressionsPerfect Square Trinomial

x2 + 6x + 9 If we’re using the Holy Grail Algorithm:

a = 1 b = 6 c = 9

So we’re looking for two numbers that multiply to get 9 (1 • 9) and add together to get 6

Those numbers have to be 3 & 3 A perfect square trinomial occurs when the numbers

are the same.

5-4: Factoring Quadratic ExpressionsFactoring: x2 + 6x + 9

x2 + 6x + 9

+

+3 +3x2 x x + 9x(x + 3) +3(x + 3)

(x + 3)(x + 3)written as(x + 3)2


16x2 – 56x + 49

The shortcut: Take the square root of the left term: Take the square root of the right term: The sign both terms share will be the sign of the

middle term:

4x7

(4x – 7)(4x – 7) = (4x – 7)2


Pg. 26437 – 45 (all problems)No work = no credit

essential question: how is foil related to factoring?

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