Factoring Trinomials of the Form ax2 + bx + c, a = 1

2 + bx + c BY GROUPING (the a • c Method) Step 1: Look for a GCF and factor it out first Step 2: Multiply the coefficient of the leading term a by the constant term c List the factors of this product (a • c) to find the pair of factors, f 1 and f 2, that sums to b, the coefficient of the middle term

Factoring Trinomials of the Form ax2 + bx + c, a = 1

Factoring Trinomials of the Form ax2 + bx + c, a = 1 When trinomials factor, the resulting terms are binomials Trinomials in the form of ax2 + bx + c where a = 1 will fall into one of three patterns for factoring Pattern 1: ax2 + bx + c In this pattern, the coefficient a is positive and both of the operators are addition This will result

FACTORING TRINOMIALS OF THE FORM: AX2 + BX + C

Put in standard quadratic form: Ax2 + Bx + C Factor out a GCF if possible Step 1 Find A•C (Ax2 + Bx + C) Step 2 Find 2 factors of the product A•C whose algebraic sum is B Same sign binomials--correct factors must ADD to B Different signs--correct factors must have a DIFFERENCE of B Step 3 Re-write middle term as the algebraic sum of the two

8th Grade

Factoring ax2 + bx + c A-SSE A 1a Interpret parts of an expression, such as Obiective To factor trinomials of the form ax2 + bx + c X+3 Getting Ready An array of three rectangular solar panels has area 3x2 + 21x + 36 The height of the array is x + 3 What is the length of the array? You did this for Explain your reasoning one panel in

8-6 ax^2+bx+c

ax2 + bx + c when a 1 different from factoring a trinomial when a I ? How is it similar? Open-Ended Find two different values that complete each expression so that the trinomial can be factored into the product of two binomials Factor your trinomi 32 + x 4 Answers may vary Sample: 19, 16: (4x + + 4); 35 8n2 4 n - 10 Answers may vary Sample:

Factoring

(continued) Form K Factoring ax2 1 bx 1 c Open-Ended Find two diff erent values that complete each expression so that the trinomial can be factored into the product of two binomials Factor your trinomials 19 4n2 1 u n 2 3 20 12r2 1 u1 6 21 24a2 1 u a 2 15 22 18b2 1 u b 1 8 23 A parallelogram has an area of 8x2 2 2x 2 45 Th e height of

2 QUADRATIC EQUATIONS

2 3 3 To Solve Quadratic Equations : ax2 + bx + c = 0 II Method of completing the square - by expressing ax2 + bx + c in the form a(x + p)2 + q [a = 1, but involving fractions when completing the square] EXAMPLE EXERCISE C3 Solve x2 – 3x – 2 = 0 by method of ‘completing the square’ Give your answer correct to 4 significant figures

Table of Basic Integrals Basic Forms

ax2 + bx+ c dx= 1 2a lnjax2+bx+cj b a p 4ac 2b2 tan 1 2ax+ b p 4ac b Integrals with Roots (17) Z p x adx= 2 3 (x a)3=2 (18) Z 1 p x a dx= 2 p x a (19) Z 1 p a x dx= 2

Table of Integrals - UMD

A - EXPRESSIONS DUN POLYNÔME DU 2nd DEGRE

P(x)=ax2+bx+c = a b2 — 4ac 4a , noté A, défini par : A = b2 — 4ac b2 — 4ac 4a2 On appelle discriminant du trinôme P le réel Forme développée ou réduite Définition 1 On appelle fonction polynôme du second degré, ou trinôme du second degré, toute fonction P définie sur R qui peut s'écrire sous la forme P(x) = ax2 + bx + c

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Factoring Trinomials of the Form ax

2 + bx + c, a = 1 When trinomials factor, the resulting terms are binomials. Trinomials in the form of ax 2 + bx + c where a = 1 will fall into one of three patterns for factoring.

Pattern 1: ax

2 + bx + c

In this pattern, the coefficient a is positive and both of the operators are addition. This will result

in the product of two monomials, both of which will have operators of addition (+). ax 2 + bx + c = (ax + n) (x + m); where n and m are factors of c

Pattern 2: ax

2 - bx + c In this pattern, the coefficient a is positive, the operator before b is subtraction (-) and the operator before c is addition (+). This will result in the product of two monomials, both of which will have operators of subtraction (-). ax 2 - bx + c = (ax - n) (x - m); where n and m are factors of c

Pattern 3: ax

± bx - c

There are two patterns shown above, either of which will give the same result. The conditions needed to fit this pattern are that a is positive a nd that the operator before c is subtraction (-). The operator before b may be either a subtraction or addition. This will result in the product of two monomials, one of which will have an operation of addition and one will have an operation of subtraction. ax 2 ± bx - c = (ax + n) (x - m); where n and m are factors of c

Steps in factoring trinomials:

1. Factor out the GCF from the trinomial (if there is one) 2. Determine which pattern applies to the given trinomial 3. Find the factors of the constant term "c" whose sum equals "b" a. For pattern 1 - both factors need to be positive b.

For pattern 2 - both factors need to be negative

c. For pattern 3 - the factors must be opposite signs 4. Use the factors from step 2 to write the trinomial in factored form a. If none of the factors work then the trinomial is "prime" and cannot be factored.

Example 1: Factor the trinomial x

2 + 5x + 6.

Solution:

Step 1: Factor out the GCF fr

om the trinomial (if there is one)

There is no GCF between the three terms of x

2 , 5x, and 6. Step 2: Determine which pattern applies to the given trinomial. The operators before the second and third terms are both addition, so this trinomial fits pattern 1. ax 2 + bx + c = (ax + n) (x + m); where n and m are factors of c Step 3: Find the factors of "c" whose sum equals "b" Since the factors must be positive the only factors of 6 are 1, 2, 3, and 6.

Factors of 6 Sum of the factors

1 × 6 = 6 1 + 6 = 7

2 × 3 = 6 2 + 3 = 5

The factors of 2 and 3 will give us the coefficient "b" in the trinomial. Step 4: Use the factors from step 3 to write the trinomial in factored form x 2 + 5x + 6 = (x + )(x + ) = (x + 2)(x + 3)

Example 2: Factor the trinomial x

2 - 7x + 12.

Solution:

Step 1: Factor out the GCF fr

om the trinomial (if there is one)

There is no GCF between the three terms of x

2 , 7x, and 12. Step 2: Determine which pattern applies to the given trinomial. The operator before the second term is subtraction and the operator before the third term is addition, so this trinomial fits pattern 2. ax 2 - bx + c = (ax - n) (x - m); where n and m are factors of c

Example 2 (Continued):

Step 3: Find the factors of "c" whose sum equals "b" Since the factors must be negative the only factors of 12 are -1, -2, -3, -4, -6, and -12.

Factors of 12 Sum of the factors

-1 × -12 = 12 -1 + -12 = -13 -2 × -6 = 12 -2 + -6 = -8 -3 × -4 = 12 -3 + -4 = -7 The factors of -3 and -4 will give us the coefficient "b" in the trinomial. Step 4: Use the factors from step 3 to write the trinomial in factored form x 2 - 7x + 12 = (x - )(x - ) = (x - 3)(x - 4)

Example 3: Factor the trinomial x

2 - 10x - 24.

Solution:

Step 1: Factor out the GCF fr

om the trinomial (if there is one)

There is no GCF between the three terms of x

2 , 10x, and 24. Step 2: Determine which pattern applies to the given trinomial.

The operators before the second and thir

d term are subtraction, so this trinomial fits pattern 3. ax 2 - bx - c = (ax + n) (x - m); where n and m are factors of c Step 3: Find the factors of "c" whose sum equals "b" Since the factors must be opposite signs the factors of 24 are ±1, ±2, ±3, ±4, ±6,

±8, ±12, and ±24.

Factors of -24 Sum of the factors

-1 × 24 = -24 -1 + 24 = 23 -2 × 12 = -24 -2 + 12 = 10 -3 × 8 = -24 -3 + 8 = 5 -4 × 6 = -24 -4 + 6 = 2

1 × -24 = -24 1 + -24 = -23 0DWK6WXGHQW/HDUQLQJ$VVLVWDQFH&HQWHU6DQ$QWRQLR&ROOHJH

Example 3 (Continued):

Factors of -24 Sum of the factors

2 × -12 = -24 2 + -12 = -10

3 × -8 = -24 3 + -8 = -5

4 × -6 = -24 4 + -6 = -2

The factors of 2 and -12 will give us the coefficient "b" in the trinomial. Step 4: Use the factors from step 3 to write the trinomial in factored form x 2 - 10x - 24 = (x + )(x - ) = (x + 2)(x - 12)

[PDF] Factoring Trinomials of the Form ax2 + bx + c, a = 1

Factoring Trinomials of the Form ax

Pattern 1: ax

Pattern 2: ax

Pattern 3: ax

± bx - c

Steps in factoring trinomials:

For pattern 2 - both factors need to be negative

Example 1: Factor the trinomial x

Solution:

Step 1: Factor out the GCF fr

There is no GCF between the three terms of x

Factors of 6 Sum of the factors

1 × 6 = 6 1 + 6 = 7

2 × 3 = 6 2 + 3 = 5

Example 2: Factor the trinomial x

Solution:

Step 1: Factor out the GCF fr

There is no GCF between the three terms of x

Example 2 (Continued):

Factors of 12 Sum of the factors

Example 3: Factor the trinomial x

Solution:

Step 1: Factor out the GCF fr

There is no GCF between the three terms of x

The operators before the second and thir

±8, ±12, and ±24.

Factors of -24 Sum of the factors

1 × -24 = -24 1 + -24 = -23 0DWK6WXGHQW/HDUQLQJ$VVLVWDQFH&HQWHU6DQ$QWRQLR&ROOHJH

Example 3 (Continued):

Factors of -24 Sum of the factors

2 × -12 = -24 2 + -12 = -10

3 × -8 = -24 3 + -8 = -5

4 × -6 = -24 4 + -6 = -2

Example 4: Factor the trinomial 2x

Solution:

Step 1: Factor out the GCF fr

4x = 2

× x

6 = 2 × 3

GCF: 2