Factoring Quadratic Expressions with Leading Coefficient 1: Master the Technique

Positive Factors of $12$	Sum
$1$ and $12$	$1 + 12 = 13$
$2$ and $6$	$2 + 6 = 8$
$3$ and $4$	$3 + 4 = 7$

Positive Factors of

12

Sum

1

and

12

1 + 12 = 13

2

and

6

2 + 6 = 8

3

and

4

3 + 4 = 7

Positive Factors of $42$	Sum
$1$ and $42$	$1 + 42 = 43$
$2$ and $21$	$2 + 21 = 23$
$3$ and $14$	$3 + 14 = 17$
$6$ and $7$	$6 + 7 = 13$

Positive Factors of

42

Sum

1

and

42

1 + 42 = 43

2

and

21

2 + 21 = 23

3

and

14

3 + 14 = 17

6

and

7

6 + 7 = 13

Positive Factors of $- 10$	Sum
$- 1$ and $10$	$- 1 + 10 = 9$
$1$ and $- 10$	$1 + (- 10) = - 9$
$- 2$ and $5$	$- 2 + 5 = 3$
$2$ and $- 5$	$2 + (- 5) = - 3$

Positive Factors of

- 10

Sum

- 1

and

10

- 1 + 10 = 9

1

and

- 10

1 + (- 10) = - 9

- 2

and

5

- 2 + 5 = 3

2

and

- 5

2 + (- 5) = - 3

Factors of $- 90$	Sum of Factors
$- 90$ and $1$	$- 90 + 1 = - 89$
$- 45$ and $2$	$- 45 + 2 = - 43$
$- 30$ and $3$	$- 30 + 3 = - 27$
$- 18$ and $5$	$- 18 + 5 = - 13$
$- 15$ and $6$	$- 15 + 6 = - 9$

Factors of

- 90

Sum of Factors

- 90

and

1

- 90 + 1 = - 89

- 45

and

2

- 45 + 2 = - 43

- 30

and

3

- 30 + 3 = - 27

- 18

and

5

- 18 + 5 = - 13

- 15

and

6

- 15 + 6 = - 9

Negative Factors of $60$
$- 1$ and $- 60$
$- 2$ and $- 30$
$- 3$ and $- 20$
$- 4$ and $- 15$
$- 5$ and $- 12$
$- 6$ and $- 10$

Negative Factors of

60

- 1

and

- 60

- 2

and

- 30

- 3

and

- 20

- 4

and

- 15

- 5

and

- 12

- 6

and

- 10

Factors of $60$	Sum
$- 1$ and $- 60$	$- 1 + (- 60) = - 61$
$- 2$ and $- 30$	$- 2 + (- 30) = - 32$
$- 3$ and $- 20$	$- 3 + (- 20) = - 23$
$- 4$ and $- 15$	$- 4 + (- 15) = - 19$
$- 5$ and $- 12$	$- 5 + (- 12) = - 17$
$- 6$ and $- 10$	$- 6 + (- 10) = - 16$

Factors of

60

Sum

- 1

and

- 60

- 1 + (- 60) = - 61

- 2

and

- 30

- 2 + (- 30) = - 32

- 3

and

- 20

- 3 + (- 20) = - 23

- 4

and

- 15

- 4 + (- 15) = - 19

- 5

and

- 12

- 5 + (- 12) = - 17

- 6

and

- 10

- 6 + (- 10) = - 16

Factors of $- 12250$	Sum of Factors
$1225$ and $- 10$	$1225 + (- 10) = 1215$
$490$ and $- 25$	$490 + (- 25) = 465$
$350$ and $- 35$	$350 + (- 35) = 315$
$245$ and $- 50$	$245 + (- 50) = 195$
$175$ and $- 70$	$175 + (- 70) = 105$

Factors of

- 12250

Sum of Factors

1225

and

- 10

1225 + (- 10) = 1215

490

and

- 25

490 + (- 25) = 465

350

and

- 35

350 + (- 35) = 315

245

and

- 50

245 + (- 50) = 195

175

and

- 70

175 + (- 70) = 105

	Expression	$w = 75$
Width	$w$	$75$
Length	$w + 105$	$75 + 105 = 175$

Expression

w = 75

Width

w

75

Length

w + 105

75 + 105 = 175

Factors of $28$	Sum of Factors
$- 28$ and $- 1$	$- 28 + (- 1) = - 29$
$- 14$ and $- 2$	$- 14 + (- 2) = - 16$
$- 7$ and $- 4$	$- 7 + (- 4) = - 11$

Factors of

28

Sum of Factors

- 28

and

- 1

- 28 + (- 1) = - 29

- 14

and

- 2

- 14 + (- 2) = - 16

- 7

and

- 4

- 7 + (- 4) = - 11

Factor each expression.

x^{2} + 12 x + 11

w^{2} + 10 w + 21

We want to factor a quadratic trinomial. To factor a quadratic trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term. x^2+12x+ 11 In this case, we have 11. Since this is a positive number, we need to find the factor pairs of 11 where both factors have the same sign (both positive or both negative).

Factors of 11
1 and 11
-1 and -11

Next, let's consider the coefficient of the linear term. x^2+ 12x+11 For this term, we need find a factor pair of 11 whose sum is to equal the coefficient of the linear term, 12.

Factor Pairs	Sum
1 and 11	12
-1 and -11	- 12

We found the factor pair of 11 whose sum is 12. Using these numbers, we can now rewrite the given expression in factored form. x^2+ 12x+ 11 = (x+1)(x+11)

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

We are given a trinomial with a leading coefficient of 1. To factor it, we will use the same procedure we used in Part A. Let's start by taking a look at the constant term. w^2+10w+ 21 In this case, we have 21. This is a positive number, so we need to find two integer factors with a positive product. Additionally, these integers must have the same sign (both positive or both negative) so that their product is 21.

Factors of 21
1 and 21
-1 and -21
3 and 7
-3 and -7

Next, let's consider the coefficient of the linear term. w^2+ 10w+21 For this term, we need the sum of the factor pair to equal the coefficient of the linear term, 10.

Factors	Sum of Factors
1 and 21	22
-1 and -21	-22
3 and 7	10
-3 and -7	-10

We found the factors of 21 whose sum is 10. Using these numbers, we can now rewrite the given expression in factored form. w^2+ 10w+ 21= (w+3)(w+7)

Let's check our answer by applying the Distributive Property.

The result is the same as the given expression. Therefore, we can be sure our solution is correct!

Factor each expression.

k^{2} - 5 k + 4

m^{2} - 17 m + 72

The given expression is a quadratic trinomial with a leading coefficient of 1. To factor the trinomial, we start by taking a look at the constant term. k^2-5k+ 4 In this case, we have 4. Since 4 is a positive number, we need to find a factor pairs of 4 where both factors have the same sign (both positive or both negative).

Factors of 4
1 and 4
-1 and -4
2 and 2
-2 and -2

Next, let's consider the coefficient of the linear term. k^2 - 5k+4 Considering this term, we need the factor pair whose sum is equal to the coefficient of the linear term - 5.

Factors	Sum of Factors
1 and 4	5
-1 and -4	- 5
2 and 2	4
-2 and -2	-4

We found the factors of 4 whose sum is - 5. Using these numbers, we can now rewrite the given expression in factored form. k^2 -5k+ 4 = (k-1)(k-4)

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

Let's start by taking a look at the constant term. m^2-17m+ 72 The constant term is 72, which is a positive number. We need to find two integers factors of 72 whose product is positive, meaning that they must have the same sign (both positive or both negative).

Factors of 72
1 and 72
-1 and -72
2 and 36
-2 and -36
3 and 24
-3 and -24
4 and 18
-4 and -18
6 and 12
-6 and -12
8 and 9
-8 and -9

Next, let's consider the coefficient of the linear term. m^2 - 17m+72 For this term, we need the sum of the factors to equal the coefficient of the linear term, -17.

Factors of 72	Sum of Factors
1 and 72	73
-1 and -72	-73
2 and 36	38
-2 and -36	-38
3 and 24	27
-3 and -24	-27
4 and 18	22
-4 and -18	-22
6 and 12	18
-6 and -12	-18
8 and 9	17
-8 and -9	-17

We found the factor pair of 72 whose sum is - 17. Using these numbers, we can now rewrite the given expression in factored form. m^2 - 17m+ 72 = (m-8)(m-9)

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

We obtained the given expression. We can be sure our solution is correct!

Factor each expression.

h^{2} + 3 h - 4

y^{2} + 3 y - 40

The given expression is a quadratic trinomial with a leading coefficient of 1. To factor the trinomial, we start by taking a look at the constant term. h^2+3h - 4 In this case, we have - 4. Since - 4 is a negative number, we need to find the factor pairs of - 4 where the factors have opposite signs (one positive and one negative).

Factors of 4
-1 and 4
1 and -4
-2 and 2

Next, let's consider the coefficient of the linear term. h^2 + 3h- 4 We need the factor pair whose sum is equal to the coefficient of the linear term, 3.

Factors	Sum of Factors
1 and - 4	- 3
- 1 and 4	3
- 2 and 2	0

We found the factor pair of - 4 whose sum is 3. Using these numbers, we can now rewrite the given expression in factored form. h^2 + 3h - 4= (h-1)(h+4)

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

We have another quadratic expression with a leading coefficient of 1. Let's start by taking a look at the constant term. y^2+3y - 40 The constant term c is - 40, which is a negative number. We now need to find two integer factors of -40. For the product of the integers to be negative, they must have opposite signs (one positive and one negative).

Factors of - 40
-1 and 40
1 and -40
-2 and 20
2 and -20
-4 and 10
4 and -10
-5 and 8
5 and -8

Next, let's consider the coefficient of the linear term. y^2+ 3y - 40 For this term, we need the sum of the factor pair to equal the coefficient of the linear term, 3.

Factors	Sum of Factors
-1 and 40	39
1 and -40	-39
-2 and 20	18
2 and -20	-18
-4 and 10	6
4 and -10	-6
-5 and 8	3
5 and -8	-3

We found the factors of - 40 whose sum is 3. Using these numbers, we can now rewrite the given expression in factored form. y^2 + 3y - 40 = (y-5)(y+8)

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

The result is the same as the given expression. Our solution is correct!

A carpet weaver weaves a rectangular rug.

The area of a rectangular rug is given by the trinomial.

r^{2} - 11 r + 18

What are the possible dimensions of the rug? Write the answer as a product of the dimensions.

The area of a particular rectangular rug is given by the following trinomial. r^2-11r+18 We see that it is a quadratic trinomial with a leading coefficient of 1. Since the trinomial represents the area of the rug, it can be expressed as a product of binomials, (r+p) and (r+q) for some integers p and q. To factor the expression, we first need to identify the coefficient of r and the constant — the values of b and c, respectively. r^2-11r+18 ⇓ r^2+( - 11)r+ 18 In this case, b= - 11 and c= 18.

Since c is positive, the factors p and q must have the same sign so that their product is positive.
Since b is negative, both p and q must be negative so that their sum is negative.

Let's now list the factor pairs of 18 where both factors are negative and identify a pair that adds up to - 11.

Factors of 18	Sum of factors
- 1 and - 18	- 19
- 2 and - 9	- 11
- 3 and - 6	- 9

The factors - 2 and - 9 are what we need. Using these factors, we can write a factored form of the given trinomial. r^2-11r+18 = (r-2)(r-9) Therefore, the possible dimensions of the rug are r-2 and r-9.

Suppose a quadratic trinomial

x^{2} + b x + c

is factored as follows for some integers

p

and

q .

x^{2} + b x + c = (x + p) (x + q)

Which statement about

p

and

q

is true when

c > 0 ?

A . B . C . D . p and q have opposite signs . Either p or q is zero . p and q have the same sign . The product of p and q is less than zero .

We know that the product of the binomials (x+p) and (x+q), where p and q are integers, is equal to the quadratic trinomial x^2+bx+c. x^2 +bx+c = (x+p)(x+q) The integers p and q must satisfy two conditions.

The product of p and q is equal to c.
The sum of p and q is equal to b.

The first condition and c > 0 imply that p and q must have the same sign so that p* q is positive. Otherwise, their product would be negative. p* q = c and c> 0 ⟹ p * q >0 We do not have any information about b, the sum of the values of p and q, so we can disregard the second condition. This means that the only conclusion we can draw is that p and q have the same sign. The answer is C.

Let's check what the value of c is when we assume that the other options are true. If what we find contradicts with c>0, we can eliminate that option.

Option A

If p and q have opposite signs, their product will be negative. Because c is equal to the product, c will also be negative.

	Signs of p and q	Sign of the Product	Sign of c
Case I	p>0 and q < 0	p* q < 0	c < 0
Case II	p< 0 and q > 0	p* q < 0	c < 0

We can eliminate the option A as the result contradicts with c >0.

Option B

When either p or q equals 0, the product of the numbers equals 0. Since pq= c, c is equal to 0 by the Transitive Property of Equality.

	Value of p or q	Value of the Product	Value of c
Case I	p=0	p* q = 0	c = 0
Case II	q=0	p* q = 0	c = 0

The option B also contradicts with c>0.

Option D

Finally, we will check the last option. In the first table, we showed that the product of p and q is less than zero when they have opposite signs. In other words, when the product is less than zero, the value of c is less than zero. Therefore, the option D can be eliminated, too.

Factoring Quadratic Expressions With a Leading Coefficient of 1

Catch-Up and Review

Finding a Pair of Numbers

Factoring a Quadratic Trinomial With Leading Coefficient $1$

Proof

Factoring $x^{2} + b x + c$ When $b > 0$ and $c > 0$

Hint

Solution

Factoring $x^{2} + b x + c$ When $b > 0$ and $c < 0$

Hint

Solution

Factoring $x^{2} + b x + c$ When $b < 0$ and $c < 0$

Answer

Hint

Solution

Factoring $x^{2} + b x + c$ When $b < 0$ and $c > 0$

Hint

Solution

Practice Factoring Quadratic Trinomials

Finding the Dimensions of a Trough by Factoring

Answer

Hint

Solution

Finding a Pair of Numbers

Answer

Hint

Solution

Option A

Option B

Option D

Factoring Quadratic Expressions With a Leading Coefficient of 1

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	Each lesson is meant to take 1-2 classroom sessions